What is Computation?
(How) Does Nature Compute?
1. Quantum vs. Classical Computation
Adrian German: About five years ago at MIT Stephen Wolfram was asked a
question1 at the end of his presentation:
Question (on tape): “How does your Principle of Computational
Equivalence (PCE) explain the separation in complexity between a
classical cellular automaton (CA) and a quantum CA?”
Stephen Wolfram (answers the question on tape): “If you take the
current formalism of quantum mechanics and you say: let’s use it,
let’s just assume that we can make measurements infinitely quickly
and that the standardized formalism of quantum mechanics is an
exact description of the actual world, not some kind of an idealization, as we know that it is – because we know that in fact when we
make measurements, we have to take some small quantum degree of
freedom and amplify it to almost an infinite number of degrees of
freedom, and things like that – but let’s assume we don’t worry about
that, let’s take the idealization that, as the formalism suggests, we
just do a measurement and it happens. So then the question is: how
does what I am talking about relate to the computation that you
can get done in a quantum case vs. the classical case. And the first
thing I should say is that, ultimately, the claim of my PCE is that
in our universe it is in fact not possible to do things that are more
sophisticated than what a classical CA, for example, can do. So, if
it turned out that quantum CAs (or quantum computers of some
kind) can do more sophisticated things than any of these classical
CAs then this PCE of mine would just be wrong.”
Question (on tape): “By more sophisticated do you mean with a
better complexity then, or do you mean in a universal sense?”
1 Plays recording from http://mitworld.mit.edu/video/149/ in which the selected paragraph appears at the 1:28:28 mark. Tape starts.
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Wolfram (on tape): “So – to be more specific, one question is: can
it do more than a universal Turing machine, can it break Church’s
thesis? Another one is: can it do an NP-complete computation in
polynomial time? And on the second issue I am not so sure. But
on the first issue I certainly will claim very strongly that in our actual universe one can’t do computations that are more sophisticated
than one can get done by a standard classical Turing machine. Now,
having said that there’s already then a technical question which is:
if you allow standard idealizations of quantum mechanics, with complex amplitudes that have an arbitrary number of bits in them, and
things like that, even within the formalism can you do computations
that are more sophisticated than you can do in a standard classical
universal Turing machine? I think that people generally feel that
you probably can’t. But it would be interesting to be able to show
that. For example, the following is true: if you say that you do computations just with polynomials then there is a result from the 1930s
that says that every question you ask is decidable. This was Tarski’s
result. So, if the kinds of primitives that you have in your theory
would be only polynomial primitives then it is the case that even
using these arbitrary precision amplitudes in quantum mechanics
you couldn’t compute something more than what we can classically
compute. But as soon as you allow transcendental functions that is
no longer the case – because for example even with trigonometric
functions you can easily encode an arbitrary diophantine equation
in your continuous system. So if you allow such functions then you
can immediately do computations that cannot be done by a classical
Turing machine, and if there was a way to get quantum mechanics to
make use of those transcendental functions – which certainly doesn’t
seem impossible – then it would imply that quantum mechanics in
its usual idealization would be capable of doing computations beyond the Church’s thesis limit. Then you can ask questions about
the NP completeness level of things and, as I said, I’m less certain
on that issue. My own guess, speculation if you want, is that if you
try and eventually unravel the idealizations that are made in quantum mechanics you will find that in fact you don’t succeed in doing
things that are more sophisticated than you could do with a classical
Turing machine type system – but I don’t know that for sure!”
Adrian German: Where this answer ends, our conference starts. As quantum
computation continues to generate significant interest while the interest in NKS
grows steadily we decided to ask ourselves: Is there really a tension between the
two? If there is – how is that going to inform us? What if the conflict is only
superficial? It was then that we remembered a quote from Richard Feynman
from the 1964 Messenger Lectures at Cornell, later published as the book “The
Character of Physical Law”:
“It always bothers me that, according to the laws as we understand
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them today, it takes a computing machine an infinite number of
logical operations to figure out what goes on in no matter how tiny
a region of space, and no matter how tiny a region of time. So I have
often made the hypothesis that ultimately physics will not require
a mathematical statement, that in the end the machinery will be
revealed, and the laws will turn out to be simple, like the chequer
board with all its apparent complexities2 .”
The seed of that hypothesis may have been planted by one of our guests today:
Ed Fredkin3 who says that in spite of his extensive collaboration with Feynman
was never sure that his theories on digital physics had actually made an impact
with the legendary physicist – until he heard it in the lectures as stated above.
(Ed Fredkin, seated next to Greg Chaitin at the table, confirms: “That’s right.”)
2. The Idea of Computation
George Johnson: Well, good morning, and I want to thank you all for coming
out and also for inviting me to what’s turned out to be a really, really fascinating
conference. Friday night just after I got here I had dinner with my old friend
Douglas Hofstadter who of course lives in Bloomington and I was thus reminded
why (and how) I later became interested in computation in the abstract. I was
already very interested in computers when Doug’s book “Godel, Escher, Bach”
came out – and I remember in graduate school picking up a copy in a bookstore
on Wisconsin Ave., in Washington, DC, and just being absolutely sucked into
a vortex – and then eventually buying the book and reading it. And as I was
reading it I thought “Well, this is great, because I am a science journalist,” and
2 However, by 1981 this hypothesis seems to have lost its appeal for Feynman; in his talk
at the First Conference on the Physics of Computation, held at MIT, he observed that it
appeared to be impossible in general to simulate an evolution of a quantum system on a
classical computer in an efficient way. He proposed a basic model for a quantum computer
that would be capable of such simulations. That talk, published in 1982, has proved to be
much more influential and more often quoted than the theme paragraph of our conference. At
the same conference Tommaso Toffoli introduced the reversible Toffoli gate, which, together
with the NOT and XOR gates provides a universal set for quantum computation. In 1985
in his notable paper, Deutsch was the first to establish a solid ground for the theory of
quantum computation by introducing a fully quantum model for computation and giving the
description of a universal quantum computer. After the pioneering work of David Deutsch,
quantum computation still remained a marginal curiosity in the theory of computation until
1994, when Peter W. Shor introduced his celebrated quantum algorithms for factoring integers
and extracting discrete logarithms in polynomial time. The importance of these algorithms is
well-known, however the theory still remains far more developed than the practice.
3 Richard Feynman’s interaction with Ed Fredkin started in 1962 when Fredkin and Minsky were in Pasadena and one evening not knowing what to do with their time “sort of
invited [them]selves to Feynman’s house,” and is very well documented by Tony Hey. Twelve
years later, in 1974, Fredkin visited Caltech again, this time as a Fairchild scholar and
spent one year with Feynman discussing quantum mechanics and computer science. “They
had a wonderful year of creative arguments,” writes Hey, “and Fredkin invented Conservative Logic and the Fredkin Gate, which led to Fredkin’s billiard ball computer.” (See
http://www.cs.indiana.edu/˜dgerman/hey.pdf)
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at the time I was working for a daily newspaper in Minneapolis, “I think we
really need to do a profile of this Hofstadter person.” It was a great way to get
introduced to this topic and I had my first of many other trips to Bloomington
to meet Doug and over the years we got to know each other pretty well – in fact
he copy-edited my very first book “The Machinery of the Mind.”
And it was while writing that book that I came across a quote from one
of tonight’s guests Tommaso Toffoli that I just remembered while I was sitting
here on these sessions and I looked it up the other night and I just wanted to
use it to get us started. This was an article in 1984 in Scientific American by
Brian Hayes4 that mentions Stephen Wolfram’s early work on CAs and mentions
Norman Packard who was at the Institute for Advanced Studies (IAS) at the
time (he later went on to co-found a prediction company in my hometown of
Santa Fe and is now researching Artificial Life) and quotes5 Dr. Toffoli as saying:
“... in a sense Nature has been continually computing the ‘next state’
of the universe for billions of years; all we have to do – and actually,
all we can do – is ‘hitch a ride’ on this huge ongoing computation.”
It was reading that and things like that really got me hooked and excited about
this idea of computation as a possible explanation for the laws of physics. We
were talking about a way to get started here to have you briefly introduce
yourselves, although most of us all know who you are and why you’re important,
but just to talk in that context of how you first got hooked to this idea and the
excitement of computation in the abstract.
Gregory Chaitin: Being a kid what I was looking for was the most exciting
new idea that I could find. And there were things like Quantum Mechanics and
General Relativity that I looked at along the way and Gödel’s Incompleteness
Theorem but at some basic level what was clearly the big revolution was the
idea of computation, embodied in computer technology. Computer technology
was exciting but I was even more interested in the idea of computation as a
deep mathematical and philosophical idea. To me it was already clear then that
this was a really major new mathematical idea – the notion of computation is
like a magic wand that transforms everything you touch it with, and gives you
a different way of thinking about everything. It’s a major paradigm shift at
a technological level, at the level of applied mathematics, pure mathematics,
as well as the level of fundamental philosophy, really fundamental questions
in philosophy. And the part about the physical universe – that part was not
obvious to me at all. But if you want to discover the world by pure thought, who
cares how this world is actually built? So I said: let us design a world with pure
thought that is computational, with computation as its foundational building
block. It’s like playing God, but it would be a world that we can understand,
no? If we invent it we can understand it – whereas if we try to figure out how
4 http://bit-player.org/bph-publications/AmSci-2006-03-Hayes-reverse.pdf
5 http://www.americanscientist.org/issues/id.3479,y.0,no.,content.true,page.1,css.print/issue.aspx
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this world works it turns into metaphysics again6 . So, obviously, I’ve hitched a
ride on the most exciting wave, the biggest wave I could see coming!
Ed Fredkin: When I was in the Air Force and was stationed at Lincoln Labs
I had the good fortune to meet Marvin Minsky almost right away and not long
afterwards I met John McCarthy and I’d get lots of advice from them. I don’t
know exactly when I first thought of the idea of the Universe being a simulation
on a computer but it was at about that time. And I remember when I told
John McCarthy this idea (that had to be around 1960) he said to me something
that in our long time of talking to each other has probably said to me maybe a
hundred times: “Yes, I’ve had that same idea,” about it being a computer. And
I said “Well, what do you think of it?” and he said “Yeah, well we can look for
roundoff or truncation errors in physics...” and when he said it I immediately
thought “Oh! He thinks I’m thinking of an IBM 709 or 7090 (I guess wasn’t out
yet) in the sky...” and I was thinking of some kind of computational process...
I wasn’t thinking of roundoff error... or truncation error. When I told this idea
to Marvin he suggested that I look at cellular automata and I hadn’t heard of
cellular automata at that point so what I had to do was to find a paper by... or,
find out what I could, I remember I couldn’t find the paper that von Neumann
had done and from that point on – which was around 1960 (say, 1959 or 1960) – I
remained interested in it7 . And on one of my first experiments on the computer
I decided to find the simplest rule that is symmetrical in every possible way, so
I came up with the von Neumann neighborhood and binary cellular automata
and I thought “what function that’s symmetrical is possible?” And, as it turned
out it’s XOR. And so I programmed that one up and I was kind of amazed by
the fact that that simple rule was at least a little bit interesting.
Rob de Ruyter: I also have to go back to my youth, or maybe high-school,
when (in the time that I was in high-school, at least) there was a lot of ebullience
still around about biology and there were a number of wonderful discoveries
6 Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory
and metamathematics, in particular a new incompleteness theorem in reaction to Gödel’s
incompleteness theorem. He attended the Bronx High School of Science and City College of
New York, where he (still in his teens) developed the theories that led to his independent
discovery of Kolmogorov complexity. Chaitin has defined Chaitin’s constant Ω a real number
whose digits are equidistributed and which is sometimes informally described as an expression
of the probability that a random program will halt. Ω has the mathematical property that
it is definable but not computable. Chaitin’s early work on algorithmic information theory
paralleled the earlier work of Kolmogorov.
Chaitin also writes about philosophy, especially metaphysics and philosophy of mathematics
(particularly about epistemological matters in mathematics). In metaphysics, Chaitin claims
that algorithmic information theory is the key to solving problems in the field of biology
(obtaining a formal definition of ‘life’, its origin and evolution) and neuroscience (the problem
of consciousness and the study of the mind). Indeed, in recent writings, he defends a position
known as digital philosophy. In the epistemology of mathematics, he claims that his findings
in mathematical logic and algorithmic information theory show there are ‘mathematical facts
that are true for no reason, they’re true by accident. They are random mathematical facts’.
Chaitin proposes that mathematicians must abandon any hope of proving those mathematical
facts and adopt a quasi-empirical methodology.
7 http://www.digitalphilosophy.org/
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made and are still being made about how life on the cellular scale works. So
that was a kind of an incredibly fascinating world for me that opened as I got
interested in biology and learned ever more intricate mechanisms. At the same
time of course we learned physics – actually in Holland in high-school you learn
them both at the same time, as opposed to in this country. And in physics
you get all these beautiful laws, mathematical descriptions of nature and they
give you a real sense that you can capture nature in pure thought. Now it
was obvious then already that going from that physical picture where you can
describe everything very precisely to the workings of a biological cell that there’s
an enormous range of things that you have to cover in order to understand how
this biological cell works and we still don’t know that in terms of underlying
physical principles. At the same time at that age when you’re in high school you
also go through all kind of hormonal, etc. transformations and one of the things
you start to realize is that (a) as you look at the world you take information in
from the world outside but (b) you also have an own mind with which you can
also think and do introspection. And you know that this introspection, your
own thoughts, somehow have to reside, they have to be built out of matter –
that probably, out there somewhere, somehow, that has to happen. Then that’s
a question that has always fascinated me tremendously and still really fascinates
me which is (if you want to put it in shorthand) the question of how matter
leads to mind, the mind-matter question and everything that’s related to that.
I was lucky enough to find a place in the physics department where there
they had a biophysics group and I could study questions of mind – probably
a simple mind, the mind of an insect, or at least little bits of the mind of an
insect – and try to study them in a very, highly quantitative way that harkens
back to the nice, beautiful mathematical description that you get in physics. So
I think this tension between matter and how you describe it in mathematical
formalisms and thought – how you introspectively know what thought is, and
presumably to some extent all animals have thoughts – and thought to some
extent takes the form of computation, of course, then [that] is the kind of thing
that drives me in my work8 . And as an experimentalist I find it very pleasing
that you can do experiments where you can at least take little little slivers off
these questions and perhaps make things a little bit more clear about the mind.
Tony Leggett: I guess I am going to be “skunk” of the group, because I
actually am not convinced that a computational approach is going to solve at
least those fundamental problems of physics which I find most interesting9 . I
8 Rob de Ruyter van Steveninck is Gill Professor in the Dept. of Physics and Program in
Neural Science at Indiana University Bloomington. Fellow of the American Physical Society,
Member on the Advisory Board for the Linda and Jack Gill Center at IU he has co-authored
Spikes: Exploring the Neural Code (MIT Press, 1999) with William Bialek and others
9 Sir Anthony James Leggett, KBE, FRS (born 26 March 1938, Camberwell, London, UK),
aka Tony Leggett, is the John D. and Catherine T. MacArthur Chair and Center for Advanced Study Professor of Physics at the University of Illinois at Urbana-Champaign since
1983. Dr. Leggett is widely recognized as a world leader in the theory of low-temperature
physics, and his pioneering work on superfluidity was recognized by the 2003 Nobel Prize in
Physics. He has shaped the theoretical understanding of normal and superfluid helium liquids and strongly coupled superfluids. He set directions for research in the quantum physics
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certainly think that the sort of ideas, like some of those that we’ve discussed at
this conference, are extremely intriguing and they may well be right – I don’t
know. And one area in which I think that one could quite clearly demonstrate
that this confluence of computational science and physics has been fruitful is of
course the area of quantum information and in particular quantum computing.
Certainly what’s happened in that area is that, as I said earlier, an approach
coming from computer science gives you a completely new way of looking at
the problems, posing them as questions that you would not have thought about
otherwise. But quite interesting actually that at least in the decade from 1995 to
2005, when quantum information clearly gained way, a lot of the papers which
appeared in Physical Review Letters at that time in some sense could have easily
appeared in 1964, but they hadn’t. Why not? Because people then didn’t have
the knowledge (or intuition) to ask such particular type(s) of questions. So
certainly I think that was a very very useful and fruitful interaction.
But I think that if someone told me, for example, that such and such a
problem which can be easily posed in physics cannot be answered by a computer
with a number of bits which is greater or equal to the total number of particles
in the universe: I don’t think I would be too impressed. And my reaction would
probably be: “All right, so what?” I don’t think that’s useful, I don’t think
that nature goes around computing what it has to do – I think it just doesn’t!
But I think perhaps a little more fundamentally, one reason I’m a little skeptical
about the enterprise is that I actually think that the fundamental questions in
physics have much more to do with the interface between the description of a
physical world and our own consciousness. One of the most obvious cases is
the infamous quantum measurement problem, which I certainly do regard as
a very serious problem; and secondly the question of the arrow of time: that
we can remember the past, that the past influences the future and viceversa.
It’s something that underlines some of the most fundamental aspects of our
existence as human beings and I don’t think we understand them at all, at least
I don’t think we understand them in physical terms, and I personally find it very
difficult to imagine how a computational approach could enhance our approach
to any of these problems.
Now one of the reasons I come to conferences like this is that I hope that I
may in the future somehow be convinced. So we’ll have to wait and see.
Cristian Calude: I guess I am probably the most conventional and ordinary
person10 in this context but my interests are in computability, complexity and
randomness, and I try to understand why we can do mathematics and what
makes us capable of understanding mathematical ideas. I am also a little bit
skeptical regarding the power of quantum computing. I’ve been involved in the
last few years in a small project in dequantizing various quantum algorithms,
of macroscopic dissipative systems and use of condensed systems to test the foundations of
quantum mechanics.
10 Cristian Calude is Chair Professor at the University of Auckland, New Zealand. Founding
director of the Centre for Discrete Mathematics and Theoretical Computer Science. Member of
the Academia Europaea. Research in algorithmic information theory and quantum computing.
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i.e., constructing classical versions of quantum algorithms which are as quick
as their quantum counterparts. And I have come to I believe that quantum
computing and quantum information say much more about physics than they
could become useful tools of computation. Finally, I am very interested to
understand the nature and the power of quantum randomness, whether this
type of hybrid computation where you have your favorite PC as the source of
quantum randomness ... can get you more. Can we surpass the Turing barrier,
at least in principle, with this kind of hybrid computation or not?
Tom Toffoli: Well, I was born by a historical accident in a post office and
that may have something to do with all of this (laughter) because it was exactly
organized like Ethernet: press a key and you bring a whole lot of lines (40 miles)
from a high value to ground, so you have to listen to what you are typing and
if what you get is different from what you are typing, or what you’re expecting
to hear, then it means that someone else is on the line too. So you both stop.
And then both of you start again, at random – exactly in the spirit of Ethernet.
But anyway, when I was five, it was right after the end of the war, my mother
already had four kids and she was very busy trying to find things for us to eat
from the black market and such – so I was left off for the most part of the day
in a place on the fifth or sixth floor. Where we lived we had long balconies with
very long rails, for laundry etc. and the rail was not raised in the middle, it was
just hooked, stuck in the wall at the endpoints on the two sides. So I was like
a monkey in a cage in that balcony and I had to think and move around and
find a way to spend time in some way. So one of the things I noticed was that
as I was pulling and shaking this thing, the rope, somehow I discovered that
by going to more or less one third of its length and shaking it in a certain way,
eventually the thing would start oscillating more and more and more. And I
discovered what you could call resonance, phase locking and other things like
that – and I felt really proud about this thing, about this power unleashed into
your [tiny] hands. But I also got very scared because – as a result of my early
scientific activity – the plaster had come out at the place where the rail was
hooked in the wall and there was even plaster outside on the sidewalk if you
looked down from the sixth floor balcony where I was, so I was sure I was going
to get into trouble when my mother would come home later that day.
In any event, the point that I’d like to make is that “I cannot understand
why people cannot understand” (as Darwin wrote to a friend) “that there’s no
such thing as simply experiments – there are only experiments ‘for’ or ‘against’
a subject.” In other words you have to ask a question first, otherwise the
experiment is not a very useful thing. Now you will say: “Fine, but how are
you going to answer the question that you’re making the experiment for?” And
what we do with computation, cellular automata and so on, is that we just make
our own universe. But the one that we make, we know it, we make the rules so
the answers that we get there are precise answers. We know exactly whether
we can make life within this CA (like von Neumann) and we know whether
we can compute certain things, so it’s a world that we make so that we have
precise answers – for our limited, invented world – and that hopefully can give
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us some light on answers for the questions in the real world. And so we have
to go continually back and forth between the worlds that we know because we
have complete control [over them] and we can really answer questions and the
other worlds about which we would really like to answer questions. And my
feeling is that computation, CA and all these things that Stephen has called our
attention to are just attempts to get the best of the two worlds, essentially. The
only world we can answer questions for is the one we make, and yet we want
to answer the questions about the world in which we are – which has all of its
imperfections, and where resonance does not give you an infinite peak but still
gives you crumbled plaster on the floor near the wall and so on.
But I think that we should be able to live with one foot in one world and
the other in the other world.
Stephen Wolfram: Let’s see, the question I believe was how did one get
involved with computation, and how did one get interested in computation and
then interested in these kind of things. Well, the story for me is a fairly long one
by now: when I was a kid I was interested in physics. I think I got interested in
physics mainly because I wanted to understand how stuff works and at the time,
it was in the 1960s, roughly, physics was the most convincing kind of place to
look for an understanding of how stuff works. By the time I was 10-11 I started
reading college physics books and so on and I remember a particular moment,
when I was 12, a particular physics book that had a big influence on me: it
was a book about statistical mechanics which had on the cover this kind of a
simulated movie strip that purported to show how one would go from a gas of
hard spheres or something like that, that was very well organized, to a gas of
hard spheres that looked completely random. This was an attempt to illustrate
the second law of thermodynamics. And I thought it was a really interesting
picture, so I read the whole mathematical explanation of the book as to why it
was this way.
And the mathematical explanation to me was pretty unconvincing, not at
all in the least because this was one of those explanations which – it’s about
actually the analog of time basically – ended with a statement that said “well,
the whole explanation could be just as well run in reverse, but somehow, that
isn’t how it works.” (General laughter) Not very convincing, so I decided: all
right, I will figure out on my own what was really going on there, I would
make my own version of this movie strip. At the time I had access to a simple
computer, it was an English computer: Eliott 903, computers we used mostly to
operate tanks and such. It had eight kilowords of eighteen bit memory. It was
otherwise a fine computer and I started trying to program this thing to work out
the second law of thermodynamics. In some respects the computer was fairly
primitive especially when doing floating point operations so I tried to simplify
the model that I had for these hard sphere gas bouncing around. What I realized
many years later was that the model that I had actually come up with was a two
dimensional cellular automata system. The thing that went wrong was: (a) I
simulated this 2DCA system and (b) I found absolutely nothing interesting and
(c) I could not reproduce the second law of thermodynamics. So that was when
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I was about 12 years old. And at that point I said: maybe there was something
else going on, maybe I don’t understand this so I’m going to work in an area
where I could learn about stuff, where I think I can understand better what’s
going on. And the area that I got involved with was particle physics. And so
I got to know all sorts of stuff about particle physics and quantum field theory
and figured out lots of things that made me kind of a respectable operative in
the particle physics business at the time.
And doing particle physics something that happened was that we had to do
these complicated algebraic calculations, something that I was never particularly
good at. I also have the point of view that one should always try to figure out
the best possible tools that one can build to help one out. And even though I
wasn’t very good myself at doing things like algebra I learned to build computer
tools for doing algebra. And eventually I decided that the best way to do the
kinds of computations I needed to do was to build my own tool for doing those
computations. And so I built a system called SMP which was the precursor of
Mathematica – this was around 1981. In order to build this computer system I
had to understand at a very fundamental level (a) how to set up a wide range of
computations and (b) how to build up a language that should describe a wide
range of computations. And that required inventing these primitives that could
be used to do all sorts of practical computations. The whole enterprise worked
out rather well, and one of my conclusions from that was that I was able to
work out the primitives for setting things up into a system like SMP. And then
for a variety of reasons when I got to started thinking about basic science again
my question was: “Couldn’t I take these phenomena that one sees in nature and
kind of invent primitives that would describe how they work – in the same kind
of way that I succeeded in inventing primitives for this computer language that I
built?” I actually thought that this would be a way of getting to a certain point
in making fairly traditional physics models of things, and it was only somewhat
coincidentally that later I actually did the natural science to explore abstractly
what the computer programs that I was setting up did. And the result of that
was that I found all sorts of interesting phenomena in CAs and so on.
That got me launched on taking more seriously the idea of just using computation, computer programs, as a way to model lots of things. I think that
in terms of the conviction for example that the universe can be represented by
computation I certainly haven’t proved that yet. I’ve only become increasingly
more convinced that it’s plausible – because I did use to say just like everybody
else: “Well, all these simple programs and things: they can do this amount of
stuff but there’ll be this other thing (or things) that they can’t do.” So, for
instance, say, before I worked on the NKS book one of the things that I believed was that while I could make simple programs representing the standard
phenomena in physics, that standard phenomena – not at the level of fundamental physics, but at the level of things like how snowflakes are formed, and so
on – I somehow believed that, for example, biological systems with adaptation
and natural selection would eventually lead to a higher level of complexity that
couldn’t be captured by these very simple programs that I was looking at. And
I kind of realized as a result of working on the NKS book and so on that that’s
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just not true. That’s a whole separate discussion, but I kind of believed that
that there would be different levels of things that could and couldn’t be achieved
by simple programs. But as I worked on more and more areas, I got more and
more convinced that the richness of what’s available easily in the computational
universe is sufficiently great that it’s much less crazy to think that our actual
universe might be made in that kind of way. But we still don’t know if that’s
right until we actually have the final theory so to speak.
3. Is the Universe a Computer?
George Johnson: Very good. And now we have a fairly formal linear exercise
for you. Let me preface it by saying that early in the conference there was,
I thought, this very interesting exchange between Dr. Toffoli and Seth Lloyd,
who’s another old Santa Fe friend. And I thought it was funny that he said he
had to leave to help his kids carve jack-o-lanterns, because shortly after I met
Seth he invited me to his house in Santa Fe that he was renting and he was
carving jack-o-lanterns then as well – this was before he was married and had
children. Our other jack-o-lantern carver then was Murray Gell-Mann. I was
writing a biography of Murray at the time and he had not agreed to cooperate,
in fact he was being somehow obstructive, but somehow sitting there with a
Nobel prize winner carving jack-o-lanterns helped break the ice.
Seth of course has described his grand vision of the universe as a quantum
computer – the universe is a quantum computer, he said, and then Dr. Toffoli
[had] an interesting subtle objection and basically I think you said that it was
this idea of the computational relation between the observer and the system,
that somebody has to be looking at it. And later, actually – to skip out to
another context during the same session – when Seth made some joke about the
election and you said, quoted Stalin saying that it doesn’t matter who votes,
it’s who counts the votes, that kind of seemed to connect back to it and got
me thinking about this whole idea of endo-physics of looking at the universe
from within or from without. So what I am wondering is this: does Seth’s view
of looking at the universe not as a computer but saying that it is a computer,
implies a God’s eye view of somebody who’s watching the computation? What
is the universe computing, if it is computing? Is the universe a computer, or is
the universe like a computer– or do the two state the exact same thing?
Tom Toffoli: Are you asking anyone in particular?
George Johnson: Anyone. Please jump in.
Stephen Wolfram: I can think of a few fairly obvious things to say. You
know, there’s the question of: “What are models?” In other words, if you
say, in the traditional physics view of things: “Is the motion of the earth a
differential equation?” No, it’s not – it’s described by a differential equation,
it isn’t a differential equation. And I don’t think, maybe other people here
think differently about it, but my view of computation as the underlying thing
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in physics is that computation is a description of what happens in physics.
There’s no particular sense in which it is useful to say the universe is a computer.
It’s merely something that can be described as a computation[al process] or
something that operates according to the rules of some program.
Cristian Calude: I agree and I would add one more idea. When you have
a real phenomenon, and a model – a model typically is a simplified version of
the reality. In order to judge whether this model is useful or not you have to
get some results (using the model). So I would judge the merit of this idea
about the universe being modeled faithfully by a gigantic computer primarily
by saying: “Please tell me three important facts that this model reveals about
the universe that the classical models can’t.”
Ed Fredkin: I will tell you one in a minute. The thing about a computational
process is that we normally think of it as a bunch of bits that are evolving over
time plus an engine – the computer. The interesting thing about informational
processes (digital ones) is that they’re independent of the engine in the sense
of what the process is: any engine that is universal (and it’s hard to make
one that isn’t) can – as long as it has enough memory – exactly produce the
same informational process as any other. So, if you think that physics is an
informational process then you don’t have to worry about the design of the
engine – because the engine isn’t here. In other words, if the universe is an
informational process then the engine, if there is one, is somewhere else.
Gregory Chaitin: George you’re asking a question which is a basic philosophical question. It’s epistemology versus ontology. In other words when you say
the Universe looks like a computer, that this is a model that is helpful – that’s
an epistemological point of view, it helps us to understand, it gives us some
knowledge. But a deeper question is what is the universe really. Not just we
have a little model, that sort of helps us to understand it, to know things. So,
that is a more ambitious question! And the ancient Greeks, the pre-Socratics
had wonderful ontological ideas: the world is number, the world is this, the
world is that. And fundamental physics also wants to answer ontological questions: “What is the world really built of at the fundamental level?” So, it’s
true, we very often modestly work with models, but when you start looking
at fundamental physics and you make models for that and if a model is very
successful – you start to think [that] the model is the reality. That this is really
an ontological step forward. And I think modern philosophy doesn’t believe in
metaphysics and it certainly doesn’t believe in ontology. It’s become unfashionable. They just look at epistomological questions or language, but mathematicians and physicists we still care about the hard ontological question: If
you’re doing fundamental physics you are looking for the ultimate reality, you
are working on ontology. And a lot of the work of lots of physicists nowadays
really resembles metaphysics– when you start talking about all possible worlds,
like Max Tegmark does or many other people do, or David Deutsch for that
matter. So philosophers have become [gotten] very timid, but some of us here,
we are continuing in the tradition of the pre-Socratics.
13
Ed Fredkin: The problem – one problem – we have is the cosmogeny problem
which is the origin of the universe. We have these two sort of contradictory
facts: one is [that] we have a lot of conservation laws, in particular mass energy
is conserved and we have the observation that the universe began, it seems, with
a big bang not so long ago, you know, just thirteen or fourteen billion years
ago. Well, basically there’s a contradiction in those two statements: because if
something began and you have a conservation law where did everything come
from, or how did it come about? And also the idea that it began at some time is
problematic – the laws of physics can’t help us right there because, if you think
a lot about of those details it’s not so much why matter suddenly appeared but
why is there physics and why this physics and stuff like that.
And there is a way11 to wave your hands, and come up with a kind of answer
that isn’t very satisfying. Which is that you have to imagine that there is some
other place – I just call it ‘other’ – and in this other place for whatever reason
there is an engine and that engine runs an informational process. And one can
actually come to some kind of feeble conclusions about this other place, because
there are some numbers that we can state about how big that computer must
be. In other words if we ask the question what would it take to run a computer
that emulated our universe, well – we can guess some quantitative numbers if
we can figure out a few things, and so you can make a few statements [educated
guesses] about that other kind of place. The point about ‘other’ is that it is
a place that doesn’t need to have conservation laws, it is a place that doesn’t
need to have concepts such as ‘beginnings’ and ‘ends’ – so there aren’t that
many constraints. And one of the wonderful things about computation is that
it is one of the least demanding concepts, if you say: well, what do you need
in order to have a computational engine? Well, you need a space of some kind.
What kind of space? How many dimensions does it have to have. Well, it
could have three, it could have two, it could have one, it could have seven –
it doesn’t matter. They can all do the same computations. Of course, if you
have a one-dimensional [space] you can spend a lot of time overcoming that
handicap. But does it need the laws of physics as we know them? No, you don’t
need to have the laws of physics as we know them. In fact, the requirements are
so minimal for having a computation compared to the wonderful rich physics
we have that it’s very, very simple. I am, in fact, reminded of a science-fiction
story, by a Polish author where there’s a robot that could make everything that
started with the letter ‘n’ and in Polish, like in Russian, or in English, the word
‘nothing’ starts with an ‘n’ so someone bored said: “OK, make nothing.” And
the robot started working and where the sky had previously been white with
so many stars and galaxies just minutes earlier, it slowly started to fade away,
little by little, galaxy by galaxy. Admittedly, it’s just a science fiction story, but
the point is that one could even inquire as to what would be the motivations
to create an emulation like this? You can imagine that there is some question,
and [then] one needs to think about: what could the question be?
We can speculate about that. But the point is that this is an explanation
11 http://en.wikipedia.org/wiki/Digital
philosophy#Fredkin.27s ideas on physics
14
that says: well there’s this thing called ‘other’ that we don’t know anything
about – as opposed to all other explanations that imply that some kind of
magic happened. Well: I don’t like magic, myself.
Rob deRuyter: A couple of sentences from a biological perspective: Let’s take
a naive standpoint that there’s a world out there and that there’s a brain and this
brain needs to understand what’s happening in the world, what’s going around
and unfortunately, maybe – or fortunately for us – this brain is an engine that
is really well adapted to information processing in the savannah, or in the trees,
or wherever. I don’t think that the brain itself is a universal computational
engine – at least I don’t see it that way – but it’s a device that is extremely well
adapted to processing information that comes to our sensory organs, in from the
world that we happen to inhabit. So if we want to start thinking about more
complex things or deeper things we need to develop tools that allow (or that
help) us translate our thoughts about the phenomena that we observe in the
world. Or the other way around: just like we developed hammers and pliers, in
dealing with phenomena in the world we need to develop tools to think [about
them]. I have no idea of what the limitations that the structure of our brain are,
and that impose – I mean, there must be limitations (in the way we think) that
impose structure on those tools that we develop to help think about things. But
computation, I think, in a sense, is one of the tools [that] we tried to develop in
dealing with the world around us. And what computation allows us to do, like
mathematics, is to be able to develop long chains of reasoning that we normally
(don’t use, I think) but that we can extend to reason about very long series,
sequences of complicated observations about phenomena [events] in the world
around us. So what interests me is this relationship between the way we think
– and the way we have evolved to think about the world around us – and the
things that we think about now in terms of scientific observations and [thoughts
about] the origin of the universe. To what extent does the hardware that we
carry around inform and determine the kinds of tools and computations and the
strategies that we’re using?
Tom Toffoli: I would like to give some examples to illustrate why the question
of whether the universe is a computer, is a really hard question. In some sense
it resembles the question of what is life. Let’s take for example the concept of
randomness: say you buy a random number generator and you start producing
numbers with it. First number you get out is 13, then you get 10, 17, and so on
– and then you start asking yourself about the numbers that you obtained: how
random are they? What is random? Is 10 a random number? Is 13 a random
number? How about 199999 – is it a random number? Then you realize that,
of course, randomness is not a property of the number, it is a property of
the process. You pay for a random number generator because you want to be
surprised. You want not to know what will come out. If you knew it – it would
not be random to you because it would be perfectly predictable. So we use that
term ‘a random number’ as an abbreviation for whatever [sequence] is produced
by a random number generator. I’ll give you another example: somebody tried
to trademark icons. When they were invented icons were small: 16 by 16 pixels
15
black and white, and you can draw, you know, some simple things with those 16
by 16 bits. So you may want to trademark them. And some greedy businessman
said: look I will go to the judge and I will try to I trademark the entire set of
16 by 16 pixel set (icons,) all of them, and everybody has to pay me. So now
if you are the judge and you have to decide whether you can or cannot allow
to someone the right to trademark not just one icon, but all of the icons that
can be made that way. And if you are the judge you really have two ways: you
say (a) either you have to give me a reason why this icon is really something
interesting or (b) you pay 5 cents for each one of the icons that you register
and being that there are 256 items you know, you don’t have to exercise any
judgment, it would just turn into a big contribution to the comunity.
In other words, the question is: what makes an icon what it is? Is it the
fact that it is 16 by 16 bits or that you have reason to believe that there is
something useful [in it]? Brian Hayes, whom you mentioned a moment ago,
once said: “What surprises me is that most people don’t use the computer
for what makes it unique and powerful – which is that it is a programmable
machine.” My partial definition of a computer is: something that can compute
(evaluate) a lot of different functions. If it can evaluate just one function, then I
wouldn’t call it a computer I would call it a special purpose machine or whatever
it is. So we may get the surprise that if we find the formula that gives us the
universe as a computer, then at that very point the universe itself becomes a
special purpose machine. I mean, we know the formula[, we know the machine,]
we know the initial conditions, and we just go: tick, tick, tick, tick. And it’s the
largest computer but nobody can program it – if this is the universe, we cannot
program it because we are inside12 of it.
So, the definition of a computer is a bit like the definition of life and the
definition of evolution or being adaptive: if there isn’t a component of adaptiveness, I wouldn’t call the thing a computer. Now the thing can be formalized
better, I just said it in an intuitive way, but I’m asking some of these questions
to try to clarify a bit what exactly it was that we wanted.
Gregory Chaitin: Look, I’d like to be aggresive about this – the best way for
me to think about something is to make claims that are much too strong (at
least it brings out the idea). So the universe has to be a computer, as Stephen
said, because the only way to understand something is to program it. I myself
use the same paradigm. Every time I try to understand something the way I
do it, is: I write a computer program. So the only possible working model of
the universe has to be a computer – a computational model. I say a working
model because that’s the only way we can understand something: by writing
a program, and getting it to work and debugging it. And then trying to run
it on examples and such. So you say that you understand something only if
you can program it. Now what if the universe decides however that it’s not a
12 First off the universe is calculating all possible states, the easiest way to see this is to
consider Everett’s interpretation of quantum mechanics. Furthermore a nuclear explosion is
a very crude way of programming the universe from the inside, any explosion is. Programs
can modify themselves too, erase themselves from memory. Wolfram’s concept of free will.
16
– that you can’t do a computational model about it. Well, then: no problem.
It just means we used the wrong computers. You know, if this universe is more
powerful than a computer model of it can be, that means that our notion of
what a computer is is wrong and we just need a notion of computer that is more
powerful, and then things are in sync. And by the way, there is a way to define
the randomness of individual numbers based on a computer (we hear: infinite
ones, from Toffoli) well, anyway, but that’s another issue13 .
Stephen Wolfram14 : One point to make, in relation to using computation
as a model of a universe: we’re used to a particular thing happening when we
do modeling, we’re used to models being idealizations of things. We say we’re
going to have a model of a snowflake or a brain or something like that, we don’t
imagine that we’re going to make a perfect model! It’s a very unusual case that
we’re dealing with in modeling fundamental physics (perhaps modeling isn’t
the right term, because what we imagine is that we’re going to actually have
a precise model that reproduces our actual universe in every detail.) It’s not
the same kind of thing as has been the tradition of modeling in natural science,
it’s much more. So when you say, when you talk about what runs it and so
on, it’s much more like talking about mathematics: you wouldn’t ask when you
think about a mathematical result, [if you] work out some results in number
theory, for example, one wouldn’t be asking all the time “Well, what’s running
all these numbers?” It’s just not a question that comes up when one is dealing
with something where [what you have] is a precise model of things.
One other point to make regarding the question of to what extent our efforts
of modeling relate to what our brains are good at doing and so on. One of the
things I am curious about is: if it turns out to be the case that we can find a
precise representation, a new representation (better word than model) for our
universe in terms of a simple program and we find that it’s, you know, program
number such and such – what do we conclude in that moment? It’s a funny
scientific situation, it kinds of reminds one of a couple of previous scientific
situations, like for instance Newton was talking about working out the orbits of
planets and so on and made the statement that once the planets are put in their
orbits then we can use the laws of gravity and so on to work out what would
happen – but he couldn’t imagine what would have set the planets originally in
motion in their orbit in the first place. So he said, “Well, the Hand of God must
have originally put the planets in motion. And we can only with our science
figure out what happens after that.” And it’s the same with Darwin’s theories:
once we have life happening then natural selection will lead us inexorably to all
the things that we see in biology. But how to cause life in the first place – he
couldn’t imagine. So some of the things I’d be curious about would be: if in
fact we do come up with a precise representation of the universe as a simple
program – what do we do then and can we imagine what kind of a conclusion we
13 Not to confuse the number with its representation: 3100 looks random in base 2 but not
very random in base 3. Same number, different representations.
14 Tony Leggett has been signaling and waiting for the microphone for some time now,
Gerardo points out. Stephen starts, Tony Leggett to follow.
17
can come to, about why this program and not another program and so on? So
one of the possibilities would be that we find out that it’s, you know, program
number 1074 or whatever it is. The fact that it is such a small number might be
a consequence of the fact that our brains are set up because they are made from
this universe in such a way that it is inevitable, and in [all] the enumerations
that we [might] use our universe will turn out to be a small number universe.
I don’t think that’s the case – but that’s one of those self fulfilling prophecies:
because we exist in our universe our universe will have to have laws that will
somehow seem simple [and intuitive] to us. I think it’s more clear than that,
but that’s one of the potential resolutions of this question: so now we have our
representation of the universe, what do we conclude metaphysically from the
fact that it is this particular universe [representation] and not the other one.
George Johnson: Dr. Leggett?
Anthony J. Leggett15 : I think with regard to the strong and forceful statement that could be related to Seth Lloyd’s argument namely that the universe
is a computer, I just have a little very naive and simple question: it seems to
me that if a statement is called for then the converse is not called for. So my
question to Seth Lloyd is: “What would it be like for the universe not to be
a computer?” And so far I fear I don’t find that particular statement truly
helpful. I do find quite helpful the thesis that it may be useful to look at the
universe in the particular framework of computational science and to ask different questions about [it, although] I have to say that I’m not yet convinced
that looking at it in this way is going to help us to answer some very obvious
questions, some of which [are usually met with a] certain amount of friction,
namely is the anthropic principle physically meaningful? That is, why do the
constants16 of nature as we know them have the particular values[/relevance]
that they have? Is it perhaps for some arbitrary reason, or maybe for some
other deeper reason. Now, of course, there have been plenty of arguments and
speculations [here] about [all sorts of things but] as far as I can see [they don’t
seem to give any] direct relationship (of the universe with a specific computational model, or the universe as a computer.) But I would really like to hear a
plausible argument as to why this point of view takes us further on these types
of questions [that I just mentioned.]
Ed Fredkin: I want to react a little bit to what Stephen was saying. There exist
areas where we use computers to write programs that are exact models, exact
and perfect in every possible way – and that is when you design a program
to emulate another computer. This is done all the time both for writing a
trace program [map and] for debugging, where you write an emulator for the
computer that the software is running on. Or you want to run software that is
for another computer like the Mac did when it switched CPUs from the 68000
to the PowerPC: they made an emulator, which is an exact implementor of the
software on another computer.
15 Has
been waiting patiently since mid-Chaitin or so.
principle
16 http://en.wikipedia.org/wiki/Anthropic
18
This relates to something that I used to call ‘The Tyranny of Universality.’
Which is: “Gee, we can never understand the design of the computer that runs
physics since any universal computer can do it.” In other words if there’s a
digital computer running all physics of course then any computer can do it, but
then, after convincing myself that that point of view made sense a long time
later I came up with a different perspective: that if the process that runs physics
is digital and it is some kind of CA there will exist a model that’s one to one
onto in terms of how it operates. And it would probably be possible to find in
essence the simplest such model so that if some kind of experimental evidence
showed us that physics is some kind of discrete, digital [physical] process like a
CA I believe we should be able to find the exact process (or, you know, one of
a small set of processes) that implement it exactly.
4. Is the Universe Discrete or Continuous?
George Johnson: Maybe a good way to get to Tony Leggett’s question that
he raised at the end is just to ask the same question that our host Adrian
asked after we saw that brief film [that he showed us] first thing this morning
which is: is there a fundamental difference between a computational physics, or a
computational model, or emulation of the universe and quantum mechanics? Or
is there a fundamental distinction between a discrete and a continuous physics?
Does anyone have a reaction to that?
Stephen Wolfram: (speaking to George Johnson): So if you’re asking is there
some definitive test for whether the universe is somehow discrete or somehow
fundamentally continuous...
George Johnson: Yeah – if there’s a conflict that [it] could possibly be both
at a deeper level.
Stephen Wolfram: If you’re asking for that – for example in the kinds of
models that I made some effort to study, there are so many different ways to
formulate these models that this question “Is it discrete [or] is it continuous?”
becomes kind of bizarre. I mean, you could say: we’ll represent it in some
algebraic form [in which] it looks like it’s talking about these very continuous
objects – and what matters about it may yet turn out to be discrete, it may
turn out to be a discrete representation (which is much easier to deal with).
So I think that at the level of models that I consider plausible the distinction
between continuity and discreteness is much less clear than we expect. I mean,
if you ask this question I think you end up asking ‘very non-physics questions’
like, for example, how much information can in principle be in this volume
of space. I’m not sure that without operationalizing that question that it’s a
terribly interesting or meaningful question.
Tom Toffoli: I would like to say something that will be very brief. Look at
CAs, they seem to be a paradigm for discreteness. But as it turns out one of
the characterizations of CAs is that they are a dynamical system [that perform
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certain kinds of translations] and they are continuous with respect to a certain
topology which is identical to the Cantor set topology, continuous in exactly that
very sense of that definition of continuity that is studied in freshman calculus.
But the interesting thing is that this is the Cantor set topology, invented by
Cantor (the one with the interval where you remove the middle third, etc.)
And as it turns out, this topology for CAs is – not in the sense of geometrical
topology, but in the sense of set topology of circuits with gates that have a finite
number of inputs and a finite number of outcomes [outputs] – that is exactly the
topology of the Cantor set, so it’s sort of a universal topology for computation.
And so we come full circle that (a) something that was not invented by Cantor to
describe computers in fact represents the natural topology to describe discrete
computers and (b) the moment you take on an in[de]finite lattice then you have
continuity exactly the kind of continuity, continuity of state, of the dynamics
that you get when you study continuous functions. So these are some of the
surprises that one gets by working on things.
Stephen Wolfram: I just want to say with respect to the question of how do
we tell, you know, the sort of thing that Tony is saying: “How do we tell that
this is [not] all just complete nonsense?” Right?
Tony Leggett: Refutation of the negative statement.
Stephen Wolfram: Yes, right. We’re really only going to know for sure if
and when we finally get a theory, a representation that is the universe and that
can be represented conveniently in computational form. Then people will say:
“Great! This computational idea is right, it was obvious all along, everybody’s
thought about it for millions of years...”
Tom Toffoli: At that point they will probably say: “In fact it’s trivial!”
Stephen Wolfram (agrees laughing, everybody laughs): And I think that until
that time one could argue back and forth forever about what’s more plausible
than what and it’s always going to be difficult to decide it based on just that.
Yet these things tend to be decided in science in a surprisingly sociological
way. For example the fact that people would seriously imagine that aspects
of string theory should be taken seriously as ways to [model the reality of a
physical universe] it’s – it’s interesting and it’s great mathematics – but it’s
a[n interesting] sociological phenomenon that causes [or forces] that to be taken
seriously at the expense of other kinds of approaches. And it’s a matter of
history that the approach we’re using (computational ideas) isn’t the dominant
theme in thinking about physics right now. I think it’s purely a matter of
history. It could be that in place of string theory people could be studying all
kinds of bizarre CAs or network systems or whatever else and weaving the same
kind of elaborate mathematical type web that’s been done in string theory and
be as convinced as the string theorists are that they’re on to the right thing.
I think at this stage until one has the definitive answer, one simply doesn’t
know enough to be able to say anything with certainty and it’s really a purely
sociological thing whether we can say that this is the right direction or this isn’t
20
the right direction. It’s very similar actually to the AI type thing, people will
argue forever and ever about whether it’s possible to have an AI and so on –
and some of us are actually putting a lot of effort into trying to do practical
things that might be identified as relevant to that. I think that actually the AI
question is harder to decide than the physics question. Because in the physics
case once we’ll have it it’s likely (it seems to me) that we’ll be able to show that
the representation is of the universe that is obviously the actual universe and
the question will be closed. Whereas the question of AI will be harder to close.
Cristian Calude: Apparently there is an antagonism between the discrete and
continuous view. But if we look at mathematics there are mathematical universes in which discrete and continuous are co-existing. Of course, what Tom
said, the Cantor space is a very interesting example, but it might be too simple
for the problem that we are discussing. For instance, non-standard analysis is
another universe where you find [this same phenomenon] and you have discreteness and you have continuity and maybe, to the extents that mathematics can
say something about the physical universe, it could be just a blend of continuous and discreteness and some phenomena may be revealed through discreteness
and some others will be revealed through continuity and continuous functions.
Gregory Chaitin: Again I am going to exaggerate – on purpose. I think the
question is like this: discreteness vs. continuity. And I’m going to say why I
am on the side of discreteness.
The reason is this: I want the world to be comprehensible! Now there are
various ways of saying this. One way would be: God would not create a world
that we couldn’t understand. Or everything happens for a reason (the principle
of sufficient reason). And other ways. So I guess I qualify as a neo-Pythagorean
because I think the world is more beautiful, [if it] is more comprehensible. We
are thinkers, we are rationalists – we’re not mystics. A mystic is a person that
gets in a communion with an incomprehensible world and feels some kind of
[comm]unity and is able to relate to it. But we want to understand rationally so
the best universe is one that can be completely understood and if the universe
is discrete we can understand it – it seems to me. This is something that you
said, at one point, Ed – it is absolutely totally understandable because you run
the model and the model is exactly what is happening.
Now a universe which uses continuity is a universe where no equation is
exact, right? Because we only have approximations up to a certain order. So
I would also say: a universe would be more beautiful if it were discrete! And
although we now end up in aesthetics, which is even more complicated, I would
still say that a discrete universe is more beautiful, a greater work of art for
God to create – and I’m not religious, by the way. But I think it’s a very good
metaphor to use – or maybe I am religious in some sense, who knows?
Another way to put it is let’s say this universe does have continuity and
messy infinite precision and everything – well, too bad for it. Why didn’t God
create as beautiful a universe as he could have?
Tom Toffoli: He should have asked you, Greg!
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Gregory Chaitin (laughs, everybody laugs): What? ... No... No ... maybe
at this point I think Stephen is the leading candidate for coming up with a ...
[everybody is still laughing, including Chaitin who continues] ... so that would
be more beautiful it seems to me. You see, it would be more understandable it
would be more rational it would show the power of reason. Now maybe reason
is a mistake as may be to postulate that the universe is comprehensible – either
as a fundamental postulate or because [you know] God is perfect and good and
would not create such a universe, if you want to take an ancient theological view.
Maybe it’s all a mistake, but this one of the reasons that I’m a neo-Pythagorean,
because I think that would be a more beautiful, or comprehensible universe.
Stephen Wolfram: I have a more pragmatic point of view, which is that if the
universe is something that can be represented by [something like] a simple discrete program, then it’s realistic to believe that we can just find it by searching
for it. And it would be embarrassing if the universe would indeed be out there
in the first, you know, billion universes that we can find by enumeration and
we never bothered to even look for it. [There’s a very sustained reaction from
the rest of the round table members, especially Greg Chaitin, whom we hear
laughing.] It may turn out that, you know, the universe isn’t findable that way
– but we haven’t excluded that yet! And that’s the stage we’re at, right now.
Maybe in, you know, 10-20-50 years we will be able to say: yes we looked at all
the first I don’t know how many – it will be like looking for counterexamples
of the Riemann hypothesis, or something like that – and we’ll say that we’ve
looked at the first quadrillion possible universes and none of them is our actual
universe, so we’re beginning to lose confidence that this approach is going to
work. But [right now] we’re not even at the basic stage of that yet.
George Johnson wants to give the microphone to Tony Leggett. Ed Fredkin
says: “Just one comment!” indicating that his comment is short.
Ed Fredkin : If this were a one dimensional universe, Steve, you would have
found the rule by now, right? Because you’ve explored all of them...
George Johnson: Tony!
Tony Leggett: Well, George, I think, raised the question whether quantum
mechanics has any relevance to this question, so let me just comment briefly on
that. I think if one thinks about the general structure of quantum mechanics,
and the ways in which we verify its predictions, you come to the conclusion that
almost all the experiments (and I’ll stick my neck out and say all the experiments) which have really shown us interesting things about quantum mechanics
do measure discrete variables in fact. Experiments on the so-called macroscopic
quantum coherence, experiments on Bell’s theorem, and so forth – they all basically use discrete variables in practice. Now of course the formalism of quantum
mechanics is a continuous formalism. You allow amplitudes to have arbitrary
values, but you never really measure those things. And I think that all one
can say when one does sometimes measure things like position and momentum
which are [apparently] continuous variables – if you look at it hard you’ll see
22
that the actual operational setup is such that you are really measuring discrete
things. So measurements within the framework of quantum mechanics which
claim to be of continuous variables usually are of discrete variables. So I think
from the, as it were, the ontological point of view, one can say that quantum
mechanics does favor a discrete point of view.
George Johnson: Well, I think we’re supposed to take a break now then we’ll
jump right in. So we’ll have a fifteen minute break.
5. Is the Universe Random?
George Johnson: First, just a technical matter, I’ve been asked by the people
recording this to remind you to speak closer into the microphone. And, also, a
procedural announcement: in the last half hour which will be starting at about
11:45 we’ll take questions from the audience, so if you can – start thinking about
the questions you wanted to ask about the conference so far.
When Hector, Gerardo and I were talking about good questions that would
stimulate debate we thought that perhaps we should ask something that would
be really really basic about randomness – and the great thing about being a
journalist particularly a science journalist is that you get this license of asking
really really smart people questions about things that have been puzzling you.
And this is something that has always kind of bugged me – the SETI (Search
for ExtraTerrestrial Intelligence) where we get these signals from space which
are then analyzed by these computers, both by super computers and by SETI
at home, where you donate some CPU time on your PC, computer cycles etc.
They’re looking for some structure in what appears to be random noise. And I
was wondering we’re getting a signal that seems to be pure noise but to some
extent – as I think Tomasso Toffoli has suggested – perhaps the randomness is
only in the eye of the beholder.
If, for example we’re getting this noisy signal that just seems to be static –
how do we know we’re not getting the ten billionth and fifty seventh digit of
the expansion of 𝜋 forward? How do we know that we’re not getting line ten
trillion quatrillion stage forward of the computation of the rule 30 automata?
So I am wondering if you can help me with that.
Tom Toffoli: This is not an answer. It’s just something to capture the imagination. Suppose that people are serious about computing and they say: “Look:
you’re not using your energy efficiently because you’re letting some energy – that
has not completely degraded – out.” So they start to make better recirculators,
filters and so on and now whatever thermal energy comes out is as thermalized
as possible. Because if it’s not, they would have commited a thermodynamical
sin. But this is exactly what happens when you look at the stars. They are
just sending close to thermodynamical equilibrium a certain temperature – so
you can say well probably then this is prima facie evidence that that there are
people there computing and they are computing so efficiently that they are just
throwing away garbage, they’re not throwing away something that is still recy-
23
clable! And this could be an explanation as to why we see all these stars with
all these temperatures.
Stephen Wolfram: You know I think the question about SETI and how it
relates to the type of things we’re talking about – I think it gets us into lots
of interesting things. I used to be a big SETI enthusiast and because I’m a
practical guy I was thinking years ago about how you could make use of unused
communication satellites and use them to actually detect signals and so on. And
now I have worked on the NKS book for a long time, and thought about the
PCE and I have became a deep SETI non-enthusiast. Because what I realized
is that it goes along with statements like “the weather has a mind of its own”.
There’s this question of what would constitute – you know, when we say that
we’re looking for extra terrestrial intelligence – what actually is the abstract
version of intelligence? It’s similar to the old question about what is life, and
can we have an abstract definition of life that’s divorced from our particular
experience with life on the earth. I mean, on the earth it is pretty easy to tell
whether something – reasonably easy to tell whether something – is alive or not.
Because if it’s alive it probably has RNA it has some membranes it has all kinds
of historical detail that connects it to all the other life that we know about. But
if you say, abstractly: what is life? It is not clear what the answer is. At times,
in antiquity it was that things that can move themselves are alive. Later on it
was that things that can do thermodynamics in a different way than other things
do thermodynamics are alive. But we still – we don’t have – it’s not clear what
the abstract definition of life is divorced from the particular history. I think the
same is true with intelligence. The one thing that most people would (I think)
agree with – is that to be intelligent you must do some computation. And with
this principle of computational equivalence idea what one is saying is that there
are lots of things out there that are equivalent in the kind of computation that
they can do ...
Tom Toffoli (who is seated next to Wolfram, and has a microphone): But
you can also do computation without being intelligent! So ... it’s a similar ...
(everybody laughs)
Stephen Wolfram (replying to Toffoli): Well[, so] that’s precisely the question:
can you – what is the difference, what is the distinctive feature of intelligence?
If we look at history, it’s a very confusing picture: a famous example that I like
is when Marconi had developed radio and (he had a yacht that he used to ply
the Atlantic with, and) at one point he was in the middle of the Atlantic and
he had this radio mast – because that was the business that he was in – and
he could hear these funny sounds, you know: ... wooo ... ooooeeo ... eeooo ...
woooo ... this kind of sounds out in the middle of the Atlantic. So what do you
think he concluded that these sounds were? He concluded that they must be
radio signals from the martians! Tesla, was in fact more convinced that they
were radio signals from the martians. But, what were they in fact? They were
in fact some modes of the ionosphere on the earth, they were physical processes
that – you know, something that happens in the plasma. So the question was,
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how do you distinguish the genuinely intelligent thing, if there is some notion
of that, from the thing that is the ... the computational thing that is.
The same thing happened with pulsars, when the first pulsars were discovered. In the first days of discovery it seemed like this periodic millisecond thing
must be some extraterrestrial beacon. And then it seemed like it was too simple. We [now] think it’s too simple to be of intelligent origin. It also relates
to this question about the anthropic principle and the question of whether our
universe is somehow uniquely set up to be capable of supporting intelligence
like us. When we realize that there isn’t an abstract definition of [actual] intelligence, it is (as I think) just a matter of doing computation. Then the space
of possible universes that support something like intelligence becomes vastly
broader and we kind of realize that this notion of an anthropic principle with
all these detailed constraints – just doesn’t make much sense.
There’s so much more we can say about this, and I’ll let others do so.
George Johnson: Randomness: is it in the eye of the beholder?
Greg Chaitin: George, I suppose it would be cowardly of me not to defend
the definition of randomness that I have worked on all my life, but I think it is
more fun to say (I was defending rationalism, you know) that a world is more
understandable because it is discrete, and for that reason it is more beautiful.
But in fact I’ve spent my life, my professional life, working on a definition of
randomness and trying to find, and I think I have found, randomness in pure
mathematics which is a funny place to find something that is random. When you
say that something is random you’re saying that you can’t understand it, right?
So defining randomness is the rational mind trying to find its own limits, because
to give a rational definition [to randomness is odd] there’s something paradoxical
in being able to know that, you know, being able to define randomness, or being
able to know that something is random – because something is random [when]
it escapes ... I’m not formulating this well, [actually] improvising it, but there
are some paradoxes involved in that. The way it works out in these paradoxes is
that you can define randomness but you can’t know that something is random,
because if you could know that something is random then it wouldn’t be random.
Randomness would just be a property like any others, and it could be used
[would enable you] to classify things. But I do think that there is a definition of
randomness for individual numbers and you don’t take into account the process
by which the numbers are coming to you: you can look at individual strings of
bits – base 10 number – and you can at least mathematically say what it means
for this to be random. Now, although most numbers or most sequences of bits
are random according to this definition, the paradoxical thing about it is that
you can never be sure that one individual number is random – so I think it is
possible to define a notion of randomness which is intrinsic and structural17 and
doesn’t depend on the process from which something comes but there is a big
problem with this definition which is: it’s useless. Except to create a paradox
17 Maybe Chaitin means ‘representation of a number’ instead. 210000 does not look random
in base 2. It does look random in base 3. The number is the same. Its representation isn’t.
The representation is a process (just as Toffoli said).
25
or except to show limits to knowledge, or limits to mathematical reason. But I
think that’s fun, so that’s what I’ve been doing my whole life.
So I don’t know if this is relevant to SETI? I guess it is, because if something
looks random it then follows that it probably doesn’t come from an intelligent
source. But what if these superior beings remove redundancy from their messages? They just run it through a compression algorithm because they are
sending us enormous messages, they’re sending us all their knowledge and wisdom and philosophy everything they know in philosophy, because their star is
about to go nova, so this is an enormous text encompassing all their accomplishments of their thinking and civilization – so obviously they think any intelligent
mind would take this information and compress it, right? And the problem is,
we’re getting this LZD compressed message and we think that it’s random noise
and in fact it’s this wonderfully compact message encapsulating the wisdom and
the legacy of this great civilization?
George Johnson: Oh, but do they include the compression algorithm?
Gregory Chaitin: Well, they might think that a priori this is the only conceivable compression algorithm, that it is so simple that any intelligent being
would use this compression algorithm – I don’t know ... (people laughing)
Stephen Wolfram: I think it’s an interesting question – about SETI. For
example, if you imagine that there was a sufficiently advanced civilization that
it could move stars around, there’s an interesting kind of question: in what
configuration would the stars be moved around and how would you know that
there is evidence of intelligence moving the stars around? And there’s a nice
philosophical quote from Kant who said “if you see a nice hexagon drawn in the
sand you know that it must come from some sort of intelligent entity [that has
created it]” And I think that it’s particularly charming that now in the last few
years it’s become clear that there are these places in the world where there are
hexagonal arrangements of stones that have formed and it is now known that
there is a physical process that has causes a hexagonal arrangement of stones
to be formed. That’s sort of a charming version of this ... (Ed Fredkin starts).
Ed Fredkin: One of the poles of Saturn has this beautiful hexagon – at the
pole and we have pictures of them.
Stephen Wolfram (continues): ... right, so the question is what do you have to
see to believe that we have evidence that there was an intention, that there was
a purpose. It’s just like Gauss for example, [he] had the scheme of carving out in
the Syberian forest the picture of the Pythagorean theorem, because that would
be the thing that would reveal the intelligence. And if you look at the Earth
now a good question to ask an astronaut is: “What do you see on the Earth
that makes you know that there is some sort of a civilization?” And I know the
answer: the thing that is most obvious to the astronauts is – two things, OK?
One is: in the great salt lake in Utah [there is] a causeway that divides a region
which has one kind of algae which tend to be of orangeish color, from a region
that has another kind of algae that tend to be bluish, and there’s a straight
26
line that goes between these two colored bodies of water. It’s perfectly straight
and that is thing number one. Thing number two is in New Zealand. There’s a
perfect circle that is visible in New Zealand [from the space]. I was working on
the NKS book and I was going to write a note about this particularly thing. We
contacted them, this was before the web was as developed as it is today, so we
contacted the New Zealand Geological Survey to get some information about
this perfect circle and they said: “If you are writing a geology book (the circle
is around a volcano,) please do not write that this volcano produces this perfect
circle, because it isn’t true.” What’s actually true is that there is a national
park that was circumscribed around the volcano and it happens to be perfectly
circular and there are sheep that have grazed inside the national park but not
outside, so it’s a human produced circle. But it’s interesting to see what is there
on the Earth that sort of reveals the intelligence of its source.
And actually, just to make one further comment about randomness, and
integers – just to address this whole idea of whether there are random integers
or not, and does it matter, and how can you tell, and so on – we have a little
project called ‘integer base’ which basically is a directory of integers. And the
question is to find is the simplest program that makes each of these integers.
And it’s interesting, it’s a very pragmatical thing [project] actually trying to fill
in actual programs that make integers. We have to have some kind of metric
as to what counts as simple18 . When you use different kinds of mathematical
functions, you use different kinds of programming constructs, you actually have
to concretely decide, [measure, quantify] simplicity. And there are lots of ways,
for example: how many times does this function appear [is referenced] on the
web, that could be a criterion as to how much weight should be given; or how
long is this function’s name in Mathematica; or other kinds of criteria like that.
So it’s kind of a very concrete version of this question about random integers.
Tom Toffoli: I’d like to say something that throws out another corollary.
There’s this self-appointed guru of electronics, Don Lancaster, he’s very well
known in circles and he said something [that is] very true. He said: the worst
thing that could happen to humanity is to find an energy source that is inexhaustible and free. You know, we are all hoping that we will find something like
that, but if we found it it would be a disaster, because then the Earth would be
turned into a cinder in no time.
If you don’t have any of the current constraints it can turn very dangerous.
For example, you have a house there, and you have a mountain, and in the
afternoon you would like to have sunshine. And in the morning you would like
to have shading from the cold or whatever, so if energy is free you just take the
mountain from where it is in the morning you take it away and you plant it back
in the evening. And this is what we’re doing in essence when we’re commuting
from the suburbs to the center of Boston. You see this river of cars that is
rushing in every day, and rushing out every day, with energy that is costing
quite a bit. Imagine if it was completely free. So, again, let’s try to think up
what the answer to our question would be if we really got it and then see the
18 Like
in genetic algorithms, or genetic programming.
27
consequences of that [first].
And, again: this comment is in relation to what I said earlier about the
randomness of the stars i[f it’]s an indication of a super-intelligence, or superstupidity. Who knows, it could be [that they’re one and] the same thing?
Rob deRuyter: Just to take your question completely literally: there’s a lot
of randomness in our eyes, as we look around the world. And especially outside
in moonlight conditions there are photons flying around, but they are not that
many and we are very aware of [the fact that] information that we’re getting
into our visual system, information that we have to process in order to navigate
successfully, is of low quality and the interesting thing is that we as organisms
are used to walking around in a world that is random and we’re very conscious
of it. Yesterday I spoke about how flies cope with this – we cope with this too
and we cope with it at all levels, from adaptation of [in] photoreceptors in our
eyes to the adaptation in the computational algorithms that our brain is using
[or executes] to where you are in an environment where you are subject to large
levels of noise, because there are not that many photons around. [In that case]
you tend to move very cautiously – you don’t start running, unless maybe the
tiger is just following you, but that is a very rare situation – so, I think, in a
lot of senses we’re used to the measurements that our sensors make being more
or less random depending on how the situation is at the moment. And so as
computational engines we are very well aware of that.
Cristian Calude: I am interested in the quality of quantum randomness. We
were able to prove that quantum randomness, under some mild assumptions on
the quantum model of physics we agree on, is not computable. So this means no
Turing machine can reproduce the outcome of a quantum generated sequence of
bits (finitely many) and this gives you a weak form of relation between Greg’s
theory, Greg’s definition of algorithmic randomness and what one would consider
to be the best possible source of randomness in this this universe, i.e., quantum
randomness. And one of the things that is delicate as we are thinking and
experimenting is a way to distinguish quantum randomness from Mathematica
generated randomness. Is it possible, by using finitely many well-chosen tests,
to find a mark of this distinction you know between something that is computer
[computably] generated from something that is not generated [in that way]?
So whereas here we have some information about the source, you know, like
Tom said – we know very well that there is an asymptotic definition of the way
that these bits can be generated – it is still very difficult to account in a finite
amount of tests for that difference.
Ed Fredkin: Just a funny story about random numbers: in the early days of
computers people wanted to have random numbers for Monte Carlo simulations
and stuff like that and so a great big wonderful computer was being designed
at MIT’s Lincoln laboratory. It was the largest fastest computer in the world
called TX2 and was to have every bell and whistle possible: a display screen that
was very fancy and stuff like that. And they decided they were going to solve
the random number problem, so they included a register that always yielded a
28
random number; this was really done carefully with radioactive material and
Geiger counters, and so on. And so whenever you read this register you got a
truly random number, and they thought: “This is a great advance in random
numbers for computers!” But the experience was contrary to their expectations!
Which was [that] it turned into a great disaster and everyone ended up hating
it: no one writing a program could debug it, because it never ran the same
way twice, so ... This was a bit of an exaggeration, but [as a result] everybody
decided that the random number generators of the traditional kind, i.e., shift
register sequence generated type and so on, were much better. So that idea got
abandoned, and I don’t think it has ever reappeared.
Stephen Wolfram: Actually it has reappeared, in the current generation of
Pentium chips there’s a hardware random generator that’s based on double
Johnson noise in the resistor. But in those days programs could be run on
their own. In these days there are problems in that programs can no longer
be run on their own: they are accessing the web, they’re doing all sorts of
things, essentially producing random noise from the outside, not from quantum
mechanics but they’re producing random noise from the outside world. So the
same problem has come up again.
George Johnson: This reminds me that in the dark ages before Mathematica the RAND corporation published this huge tome – I found a reprint of it
called “One hundred thousand random numbers” (someone corrects, Calude I
think, George Johnson repeats the correction out loud: “One million random
numbers”) in case you needed some random numbers – and Murray Gell-Mann
used to tell a story that he was working at RAND and at one time, I think when
they were working on the book, they had to print an errata sheet! (There is
laughter, but Wolfram quickly wants to make a point.)
Stephen Wolfram: Well, the story behind the erratum sheet I think is interesting because those numbers were generated from (I think a triode, or something)
some vacuum tube device and the problem was that when they first generated
the numbers, they tried to do it too quickly, and basically didn’t wait for the
junk in the triode to clear out between one bit being found and the next bit
being found. This is exactly the same cause of difficulty in randomness that
you get in trying to get perfect randomness from radioactive decay! I think
the null experiment for quantum computing, one that’s perhaps interesting to
talk about here, is this question of how do you get – I mean, can you get – a
perfect sequence of random bits from a quantum device? What’s involved in
doing that? And, you know, my suspicion is the following: my suspicion would
be that every time you get a bit out you have to go from the quantum level
up to the measured classical level, you have to kind of spread the information
about this bit out in this bowl of thermodynamic soup of stuff so that you get a
definite measurement; and the contention, or my guess, would be that there is a
rate at which that spreading can happen and [...] that in the end you won’t get
out bits that are any more random than the randomness that you could have
got out just through the spreading process alone without kind of the little quan-
29
tum seed. So that’s an extreme point of view, that the extra little piece (bit)
of quantumness doesn’t really add anything to your ability to get out random
bits. I don’t know if that is correct but you know I, at least a long time ago, I
did try looking at some experiments and the typical thing that’s found is that
you try to get random bits out of a quantum system quickly and you discover
that you have 𝑓1 noise fluctuations because of correlations in the detector and
so on. So I think that the minimal question from quantum mechanics is: can
you genuinely get sort of random bits and what’s involved in doing that? What
actually happens in the devices to make that happen? I’d be curious to know
the answer to this [question].
Tom Toffoli: I know the answer, Intel already says: yes, you can get good
quantum numbers if you have a quantum generator of random numbers. Just
generate one random number, then throw away your generator and buy a new
one, because the one that you have used is alreay entangled with the one it
generated. So you buy a new one and you solved the problem. (Wolfram says:
this is also a good commercial strategy/solution ... people laugh)
Ed Fredkin: There’s a great story in the history of – back in the ’50s people
doing various electronic things needed noise. They wanted noise, random noise
so [they thought:] what would be a good source of it. And so it was discovered
that a particular model of photomultiplier, if you covered it up and let no light
into it gave beautiful random noise. And [as a result] various people conducted
experiments, they characterized this tube and it was essentially like a perfect
source of random noise, and the volume [of sales] started [to pick up]. Various
people started building these circuits and using them all over. Meanwhile, back
at the tube factory which was RCA, someone noticed: “Hey, that old [noisy]
photomultiplier that we had trouble selling lately, sales are picking up, we better
fix that design so it isn’t so noisy!” So they fixed it and that was the end of
that source of random[ness] noise.
6. Information vs. Matter
George Johnson: I think I’ll ask another question about one other thing that
has been bothering me before we start letting the audience jump in. I first ran
across this when I was writing a book called “Fire of the mind” and the subtitle
was ‘Science, faith and the search for order.’ And I was re-reading a book
that I read in college that really impressed me at the time, and seeing that it
still [stood up,] which was Robert Pirsig’s book “Zen and the art of motorcycle
maintenance” – and it did stand up in my opinion.
There’s a scene early on in the book the protagonist who called himself
Phædrus after Plato’s dialogue is taking a motorcycle trip around the country
with his son Chris – who in real life later was tragically murdered in San Francisco where he was attending a Zen monastery, which is neither here or there
– but, in the book this person’s running around with Chris and they’re sitting
around the campfire at night and drinking whisky and talking and telling ghost
30
stories and at one point Chris asks his father: “Do you believe in ghosts?”
And he says “Well, no, of course I don’t believe in ghosts because ghosts
contain no matter and no energy and so according to the laws of physics they
cannot exist.” And then he thinks for a moment and says: “Well, of course the
laws of physics are also not made of matter or energy and therefore they can’t
exist [either].” And this really seemed like an interesting idea to me, and when
I was learning about computational physics, this made me wonder: where are
the laws of physics? Do you have to be a Platonist and think that the laws of
physics are written in some theory realm? Or does this computational physics
gives us a way to think of them as being embedded within the very systems that
they explain? [waits a little, sees Chaitin wanting to answer, says:] Greg!
Gregory Chaitin: George, we have ghosts: information! Information is nonmaterial.
George Johnson: Information is physical, right?
Gregory Chaitin: No, no – it’s ...
George Johnson: Well, wouldn’t Landaurer say that?
Gregory Chaitin: Well, maybe Rolf would say that, but ontologically we’ve
come up with this new concept of information and those of us that do digital
physics somehow take information as more primary than matter. And this is a
very old philosophical debate: is the world built of spirit or mind or is it built of
matter? Which is primary which is secondary? And the traditional view of what
we see as the reality, is that everything is made of matter. But another point
of view is that the universe is an idea and therefore (and information is much
closer to that) made of spirit, and matter is a secondary phenomenon. So, as a
matter of fact, perhaps everything is ghost. If you believe in a computational
model informational model of the universe, then there is no matter! It’s just
information – patterns of information from which matter is built.
George Johnson: Does that sound right to you, Tony?
Tony Leggett: Who, me?
George Johnson: Yes. Does it sound right to you that information is more
fundamental than matter or energy? I think most of us – you know, the average
person in the street – asked about information would think about information
as a human construct that is imposed on matter or energy [that we make.] But
I really like the idea that Gregory [has] suggested – and I actually wrote about
it quite a bit in this book – that information is actually in the basement there,
and that matter and energy are somehow [built/conjured] out of that. So I was
just wondering, from your perspective: is that something [that] ...
Tom Toffoli (starts answering the question): Well, ideally, I mean...
George Johnson: Actually I was asking Tony...
Tom Toffoli: I will be very brief – will be done right away.
31
George Johnson: Sure, go ahead.
Tom Toffoli: You can ask the same thing about correlation rather than information because it conveys the same meaning. Furthermore one can bring up
and discuss the notion of entanglement, and in the same spirit: it’s not here nor
there. Where is it? Very related issues. That’s the point I wanted to make.
Tony Leggett: Well I think I would take the slightly short-sighted point of
view [chuckle] that information seems to me to be meaningless – unless it is
information about something. Then one has to ask the question: “what is the
something?” I would like to think that that something has something to do with
the matter of interest, and the matter and energy that’s [involved].
Stephen Wolfram: This question about whether abstract formal systems are
about something or not is a question that obviously has come up from mathematics. And my guess about the answer to this question: is information the
primary thing or is matter the primary thing? I think that the answer to that
question would probably end up being that they are really the same kind of
thing. That there’s no difference between [them]. That matter is merely our
way of representing to ourselves things that are in fact some pattern of information, but we can also say that matter is the primary thing and information is just
our representation of that. It makes little difference, I don’t think there’s a big
distinction – if one’s right that there’s an ultimate model for [the] representation
of universe in terms of computation.
But I think that one can ask this question about whether formal systems
are about something – this comes up in mathematics a lot, we can invent some
axioms system and then we can say: is this axiom system describing something
really, or is it merely an axiom system that allows us to make various deductions
but it’s not really about anything. And, for example, one of the important
consequences of Gödel’s theorem is that you might have thought that the Peano
axioms are really just about integers and about arithmetic but what Gödel’s
theorem shows is that these axioms also admit different [various] non-standard
arithmetics, which are things that are not really like the ordinary integers, but
still consistent with its axioms. I think it’s actually a confusion of the way in
which mathematics has been built in its axiomatic form that there is this issue
about ‘aboutness’ so to speak – and maybe this is getting kind of abstract.
But when we think about computations we set things up, we have particular
rules, [and] we just say “OK, we run the rules, you know, [and] what happens
– happens.” Mathematics doesn’t think about [these] things in those terms,
typically. Instead, it says: let’s come up with axiom systems which constrain
how things could possibly work. That’s a different thing from saying let’s just
throw down some rules and then the rules run and then things just happen.
In mathematics we say: let’s come up with axioms which sort of describe how
things have to work, same thing was done in physics with equations – the idea of,
you know, let’s make up an equation that describe what’s possible to happen
in the world – and not (as we do in computation) let’s do something where
we set up a rule and then the rule just runs. So, for example, that’s why in
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gravitation theory there’s a whole discussion about “What do the solutions to
the Einstein’s equations look like?” and “What is the set of possible solutions?”
and not (instead) “How will the thing run?” but the question traditionally asked
there is: “What are the possible things consistent with these constraints?”
So in mathematics we end up with these axiom systems, and they are trying
to sculpt things, to ensure that they’re really talking about the thing that you
originally imagine[d you were] talking about, like integers. And what we know
from Gödel’s theorem and so on, is that that kind of sculpting can never really
work. We can never really use this constraint-based model of how to understand
things to actually make our understanding be about a certain thing. So, I think
that’s kind of the ultimate problem with this idea of whether the laws that one’s
using to describe things and the things themselves are one and the same or they
are different, and if so what the distinction really is.
Gregory Chaitin: There’s this old idea that maybe the world is made of
mathematics and that the ultimate reality is mathematical. And for example
people have thought so about continuous mathematics, differential equations
and partial differential equations, and that view was monumentally successful,
so that already is a non-materialistic view of the world. Also let me say that
quantum mechanics is not a materialistic theory of the world; whatever the
Schrödinger wave equation is it’s not matter, so materialism is definitely dead
as far as I am concerned. The way Bertrand Russell put it is: if you take the view
that reality is just what the normal day appearances are, and modern science
shows that everyday reality is not the real reality, therefore – I don’t know how
he called this ... ‘naive realism’ I think – and if naive realism is true then it’s
false, therefore it’s false. That’s another way to put it. So the only thing we’re
changing in this view that the actual structure of the world is mathematical,
the only thing new that we’re adding to this now is we’re saying mathematics
is ultimately computational, or ultimately it is about information, and zeroes
and ones. And that’s a slight refinement on a view that is quite classical.
So I am saying in a way we’re not as revolutionary as it might seem, this
is just a natural evolution in an idea. In other words, this question of idealism
vs. materialism, or “Is the world built of ideas or is the world built of matter?”
it might sound crazy, but it’s the question of “Is the ultimate structure of the
world mathematical?” versus matter. That sounds less theological and more
down to earth. And we have a new version of this, as ideas keep being updated,
the current version of this idea is just: “Is it matter or is it information?” So
these are old ideas that morph with time, you know, they evolve, they recycle,
but they’re still distinguishably not that far from their origin.
Cristian Calude: Information vs. matter, discrete vs. continuous: interesting philosophical contrasts and I found Stephen’s description to be extremely
interesting, because I too think that these views should in fact coexist in a productive duality. And it depends on your own abilities, it depends on your own
problems if one of them would be more visible or useful. And at the end of
the day what really counts is – if you have a view that in a specific problem
information prevails – what can you get (from that)? Can you prove a theorem,
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can you get some result, can you build a model which you know answers an important question or not? In some cases one view may be the right one, in other
cases the other one is, so from my point of view, which is, I would guess, more
pragmatic, I would say: look, you choose whatever view you wish in order to
get a result and if you get the result that [means that] for this specific problem
that choice was the correct one.
Ed Fredkin: But in fact the world either is continuous or discrete, and we can
call it anything we want, to get results and so on, to add convenience – but there
really is an answer to it and one answer is right and the other is wrong. I go
with Kronecker who said “God invented the integers and all else is the work of
man.” So you can do anything with discrete models and/or continuous models
but that doesn’t mean that the world is both, or can be, or could be either. No,
the world is either one – or the other.
Stephen Wolfram: This whole question about mechanism [versus] mathematics and so on – it is kind of amusing. For example with CAs, models and
things like that, people in traditional physics (not so in other areas but people
in traditional physics) have often viewed this kinds of models with a great deal
of skepticism. And it’s kind of an amusing turn of historical fate, because, in
the pre-Newtonian period people always had mechanistic models for things –
whether there were angels pushing the Earth around the orbit, or other kinds
of mechanistic type of things. And then along came this kind of purely abstract mathematical description of law of gravity and so on and everybody said
– but only after a while! – everybody said: “Well, it’s all just mathematics
and there isn’t a material reality, there isn’t a mechanism behind these things!”
And so, when one comes with these computational models which seem to have
much more of a tangible mechanism, that is viewed as suspicious and kind
of non-scientific by people who spend their lives working in the mathematical
paradigm, that it can’t be simple enough if there’s an understandable mechanism to [behind] things. So it’s kind of an interesting turning around of the
historical process. As you know I work on various different kinds of things and,
as I said, I’ve not been working much on finding a fundamental theory of physics
lately. I actually find this discussion [as] an uptick in my motivation and enthusiasm to actually go and find a fundamental theory of physics. Because I think,
in a sense, what with all of these metaphysical kinds of questions about what
might be there, what might not be there and so on: damn it, we can actually
answer these things!
Tom Toffoli: I know a way out of this! It is similar to the one that Ed proposed a long time ago. He said (about computability, referring to exponential,
polynomial problems) he said that one can turn all problems into linear problems. You know, all the exponential problems! And people of course said: “Ed
you are an undisciplined amateur you say these things without knowing what
you’re talking about.” But he said: “Look, we have Moore’s law! And with
Moore’s law everything doubles its speed every so many years. So we just wait
long enough and we get the solution – as long as it takes.”
34
So the key then, is to wait, as long as we need to. With this, I am now
giving a simpler solution to this problem, [of finding a fundamental theory of
physics,] starting from the observation that domesticated animals become less
intelligent than wild animals. This has been proven recently with research
on wolves. And maybe domesticated animals are somewhat less intelligent in
certain ways but they can see what humans want and they obey. But then
some more experiments were run and the findings were that wild wolves once
they put them in an environment with humans they learn humans faster than
domesticated animals to anticipate the wills of their trainers. It’s not that they
follow their will, but they anticipate it faster.
Now we are doing a big experiment on ourselves, on humanity – humanity
is domesticating itself. And there was a time when people said: we discovered
differential equations and very few people can understand them so we have the
monopoly, we are the scientists. And eventually somebody said: “Well, wait a
second, but why can’t we find a model like, you know, Ed or Stephen – that is,
sort of computational, discrete so that everyone can own it and possess it and so
on?” But [one forgets that] we are domesticating ourselves! To the point that
the computer that – according to Brian Hayes the first thing about a computer
is that you can program it, make it do whatever we want – now most people
don’t even know that that is possible, they just think of the computer as an
appliance. And soon even the computer will be a mystery to most people! The
programmable digital computer [like] the differential equations were a generation
ago – so we solved the problem, in that sense, just wait long enough and nobody
will even [be able to] care about these things, it will just be a mystery for us19 .
Cristian Calude: Well, I would like to just add a small remark, suggested by
your idea about physical computation. So, this is essentially a personal remark
about the P vs. NP problem: and I believe this is a very challenging and deep
and interesting mathematical question, but I think one that has no computer
science meaning whatsoever. For the simple fact that P is not an adequate
model of physical computation, and there are lots of results – both theoretical
and experimental – which point out that P does not model [properly] what we
understand as physical computation. Probably the simplest example is to think
about simplex which is exponentially difficult, but works much better in practice
than all the known polynomial solutions.
7. Unmoderated Audience Questions
George Johnson: Thank you. I guess this is a good time to move on to
questions from the audience and I think the best way to do this is that if you
ask a question of a particular speaker [if] the speaker can repeat the question
we will then have a record of what was asked – like Charles Bennett’s raindrop
craters that we saw the other day – along with what was answered.
19 And
we’ll get used to it like we get used with the idea that we are not immortal!
35
Jason Cawley (a bit incomprehensible, has no microphone): So I have a question [for Greg Chaitin but for everyone else as well.] You said that the world
would be more intelligible and prettier, and rationalism [would be] morphing
into ... [... inner system is ...] discrete – which was very [attractive] to me. But
how intelligible would it be, really, even if I grant you finiteness and discreteness,
even if we find the rule? Won’t it have all these pockets of complexity in it,
wouldn’t it have [...] computation, won’t it be huge compared to us, [which are]
finite, wouldn’t it last much longer than us – and then we still have all kinds of
ways that would be mysterious to us in all the little detail?
George Johnson asks Gregory Chaitin to repeat the question.
Gregory Chaitin: Well, I didn’t catch all of that... (collects his thoughts then
proceeds to summarize the question as best as he heard it) ... Oh, I guess the
remark is disagreeing with what I said that a discrete universe would be more
beautiful ... no, no, more comprehensible ... right? ... and you gave a lot
of reasons why you think it would be ugly, incomprehensible, disgusting – and
I can’t argue, if you feel that way! [But in that case] I don’t think that’s a
question, I view that as a comment that doesn’t need to be answered ...
[Gerardo Ortiz takes the microphone from the moderator’s table to Jason
Cawley, whom we can now hear well, reformulating his question.]
Jason Cawley: Sorry. The question is “How intelligible is a perfectly discrete
universe?” The reason I am concerned about this is: I happen to like rationalism
too, but I don’t want people concluding, when they see non intelligibility in the
universe, that it is evidence against [the] rational.
Gregory Chaitin refines his answer: Oh, okay, great! Well, in that case, as
Stephen has pointed out in his book, it could be that all the randomness in the
world is just pseudo randomness, you know, and things only look unintelligible,
but they are actually rational. The other thing is he’s also pointed out – and
this is sort of his version of Gödel’s incompleteness theorem – is that something
can be simple and discrete and yet we would not be able to prove things about
it. And Stephen’s version of this (which I think is very interesting) is that in
the way that the universe is created, because you have to run a computation, in
general you have to run a physical system to see what it will do, you can’t have
a shortcut to the answer20 . So that can be viewed as bad, but it also means
that you could have a simple theory of the world that wouldn’t help us much
to predict things. And you can also look at it as good, because it means that
the time evolution of the universe is creative and surprising, it’s actually doing
something that we couldn’t [know] in advance – by just sitting at our desks[,
and thinking] – so I view this as fundamental and creative! And in regards to
creativity Bergson was talking 100 years ago about “L’Evolution Créatrice21 ” –
at this point, this would be a new version of that. But over aestetics one can’t
20 Via
the Principle of Computational Irreducibility (which is a corollary of the PCE).
book by French philosopher Henri Bergson. Its English translation appeared in 1911.
The book provides an alternate explanation for Darwin’s mechanism of evolution.
21 1907
36
ultimately argue too much. But still, I think it was a good question.
Gerardo Ortiz: So, further questions?
New question from the audience22 : First a question about SETI: I may be
naive, but it seems strange to me why we would like to look at perfect circles and
dividing lakes when one can simply look at Chicago and New York emanating
light from Earth at night. If I were a Martian that’s what I would do. But
what interests me more and that’s the question for Sir Leggett is this: suppose
that there were phase transitions from quantum to classical. Would building
a quantum computer – would the approach to build a quantum computer be
different than trying to reduce decoherence as it’s being done presently?
Tony Leggett: I’m not entirely clear what you mean by postulating that there
was a phase transition from quantum to classical. Could you elaborate a bit?
Question is refined: Ah, well, I am a bit ignorant about your theory but, if there
is a theory that it’s not just decoherence but in fact there are phase transitions
from quantum to classical at some level – that’s why we don’t see Schroedinger’s
cat after some [... W]ould this imply perhaps a different approach of building a
quantum computer?
Tony Leggett: If you mean – if you’re referring to theories, for example, of
the GRWB type which postulate that there are physical mechanisms [systems]
which will meet the linear formalism of quantum mechanics23 which will have
to be modified – and it will have to be modified more severly as it goes from the
microscopic to the macroscopic – then I think the answer to your question is that
to the extent that we want to use macroscopic or semi-macroscopic systems as
qubits in our quantum computer, it wouldn’t work. On the other hand I don’t
think that theories of the GRWB – scenarios of the GRWB type – necessarily [...]
against an attempt to build a quantum computer always keeping the individual
bits at the microscopic level. You have to look at it in, of course, in detail in
a specific context of a particular computer built around a particular algorithm,
such as, say, Shor’s algorithm. But I don’t think it is a priori essential that a
GRWB type scenario would destroy the possibility of it [...].
Question ends, and Gerardo Ortiz invites new questions.
New question from the audience24 : Well, we still haven’t answered the question of “How does nature compute?” We are just discussing [differences] between
discrete and continuous, classical vs. quantum computation but if we see mathematics as a historical accident and we try to push it aside, shouldn’t we try
to look at nature, and try to understand how nature in fact computes? And
not just try to traduce25 it into a mathematical context. For example one can
watch plants and see that there is some kind of parallel type of computation
22 I
don’t know the name
of kets
24 Still don’t know the name, but Hector might be able to identify him.
25 I don’t know if the speaker means, literally, ‘traduce’ – or, perhaps ‘translate’ and the
meaning is [somewhat] lost in the ad-hoc translation...
23 Composed
37
that is going on, that is, not based on our mathematics, but like they do it, the
plants ... you know... Do you have you any thoughts on that?
Stephen Wolfram answers it: I think that one of the things that makes that
somewhat concrete is the question of how we should build useful computational
devices. Whether our useful computational devices should have ALUs (Arithmetic Logic Unit) inside them [or not]. The very fact that every CPU that’s
built has a piece that’s called “the arithmetic logic unit,” tells you that in our
current conception of computation we have mathematics somewhere in the middle of it. So an interesting thing is: can we achieve useful computational tasks
without ever having an ALU in the loop, so to speak. I think the answer is: definitely yes, but as the whole engineering development of computers has been so
far we’ve just been optimizing this one particular model that’s based on mathematics. And as we try to build computers that are more at the molecular scale,
we could, actually, use the same model: we could take the design of the Pentium
chip and we can shrink it down really really small and have it implemented in
atoms. But an alternative would be that we can have atoms do things that they
are more naturally good at doing. And I think the first place where this would
come up are things like algorithmic drugs, where you want to have something
that is in essence a molecule operating in some biological, biomedical context
and it wants to actually do a computation as it figures out whether to bind to
some site or not, as opposed to saying “I am the right shape so I’m going to bind
there!” So that’s a place where computation might be done. But it’s not going
to be computation that will be done through arithmetic but we’ll be forced to
think about computation at a molecular scale in its own terms, simply because
that’s the scale at which the thing has to operate. And I’m going to guess that
there will be a whole series of devices and things, that – mostly driven by the
molecular case – where we want to do computation but where the computation
doesn’t want to go through the intermediate layer of the arithmetic.
Cristian Calude: Yes! There is a lot of research that we have started about
ten years ago in Aukland, and a series of conferences called “Unconventional
Computation.” And this is one of the interesting questions. You know, there
are basically two streams of thought: one is quantum computation, and the other
one is molecular computing. And in quantum computing, you have this tendency
of using the embedded mathematics inside. But in molecular computing, you
know, you go completely wild [because] mathematics is not there [so] you use
all sorts of specific biological operations for computation. And if you look at
the results, some of them are quite spectacular.
Small refinement: Yeah, but they are still based on logic gates and ... you
know ... molecular computing is still based on trying to build logic gates and ...
Cristian Calude listens and continues: No! There is no logic gate! That’s
the difference, because the philosophy of approach in quantum computation for
instance [is:] you do these logical gates at the level of atoms or other particles –
but in molecular computation there is no arithmetical instruction, there are no
numbers, you know, everything is [a] string, and the way they are manipulated
38
is based exactly on biological type of processing. No arithmetic.
Gregory Chaitin: No boolean algebra, not [even] and ’s and or ’s?.
Cristian Calude: No, nothing! And then this is the beauty, and in a sense
this was the question that I posed to Seth Lloyd in ’98. [I said] you know, why
don’t you do in quantum computing something similar? Why don’t you try
to think of some kind of rules of processing – not imposed from the classical
computation, from Turing machines, but rules which come naturally from the
quantum processes – just like the typical approach in molecular computation.
Tom Toffoli: I would like to give a complementary answer to this. I’ve been
teaching microprocessors and microcontrollers – you have them in your watches
and cell phones. They’re extremely complicated objects. And you would say, I
mean given the ease with which we can fabricate these things, whenever we want
to run a program or algorithm, we could make a very special purpose computer
rather than using a microprocessor to program it. Apparently it turns out that
it’s much more convenient, if someone had designed a microprocessor with an
ALU and a cache and the other things in between, to just take it as a given
and then the [process is complete]. If we look at biology, biology has done the
same thing. We have, at a certain moment, hijacked mitochondria that do the
conversion of oxygen and sugar into recharging the ATP batteries. That was
a great invention. And now, after probably three billion years, or something
like that we still keep using that instead of inventing a method that is, maybe a
little more efficient, but would have to be a very special choice for every different
[circumstance]. Essentially we could optimize more, but we would do that at
the cost of losing flexibility, modularity and so on [but, apparently] it’s much
more convenient. For three billion years we’ve kept using this kind of energy
microprocessor, that worked fairly well ... So there is, essentially, the flexibility
or modular evolution that really suggests that choices like the ALU ... is not an
optimal choice, but – empirically – is a very good choice. This is my viewpoint.
Stephen Wolfram: Well – as is always the case in the history of technological
evolution it is inconceivable to go back and sort of restart [the whole design
process]. Because there is just far too much investment in that particular thing.
The point that I want to make is that – what will happen is that there will
be certain particular technological issues, which will drive different types of
computing. My guess is that the first ones that will actually be important
are these biomedical ones, because they have to operate on this molecular scale,
because that’s the scale on which biomedicine operates. And, you know, one can
get away with having much bigger devices for other purposes – but biomedicine
is a place where potentially a decision has to be made by a molecule. And
whether – maybe there will be ways of hacking around that [but] in time, it
won’t be very long before [the first applications of this kind will be finalized].
And what’s interesting about [this] is that if you look at the time from Gödel
and Turing to the time when computers became generic and everybody had
one, and the time from Crick and Watson and DNA to the time when genomics
becomes generic – it’s about the same interval of time. It hasn’t yet happened
39
for genomics, it has more or less happened for computers. Computers were
invented, the idea of computers is twenty-three years earlier or something like
that, than the DNA idea. Anyway, [it will soon happen for genomics as well,
and] in time we will routinely be able to sequence things, in real time, from
ourselves, and we’ll do all kinds of predictions that yes, you know we detect
that today you have a higher population of antibodies that have a particular
form so we’ll be able to run some simulation that this means that you should go
and by a supply of T-cells, that has this particular characteristic and so on. And
there’s a question whether the decisions about that will be made externally by
ALU based computers, or whether they will be made internally by some kind of
molecular device – more like the way biology actually does it. And if it ends up
that they are made externally then there won’t be any drive from the technology
side to make a different substructure for computing.
Roshan James: I study computer science. And one of the ideas I find truly
fascinating and I think it [really] is – is this thing called the Curry-Howard
isomorphism, which basically relates propositions to types and rules to terms
and proof normalization to program evaluation. And since you’re [discussing]
models of computation, I was wondering if you have encountered something
similar for cellular automata and such.
Stephen Wolfram: This is an area I don’t know much about, so ... [He turns
and asks Calude: “Do you know about this question?” Calude gestures, then
apparently makes a suggestion. Wolfram turns back, speaks to the student:]
OK, so you have to explain it to us a bit more. Tell us, explain to us what you’re
talking about, so we can learn something and try to address the question.
Roshan James: OK. I think that the classification, this particular relation is
very explicit in the simply typed lambda calculus where program terms, which
are lambda expressions, can be given types – and the types are pretty much
propositional logic expressions. And if you can prove a certain proposition
the structure of the proof in natural deduction style will actually look like a
program type and if the proof is not a normal proof then the process of proof
normalization is basically the process of the evaluation of the term into a normal
form of the term, which basically means that if you have this computational
model which is the lambda calculus, reductions of the calculus will correspond
to normalizations of the proofs, and the types serve as a way of classifying
programs, types are a way of saying: these particular programs that behave
in such and such a way don’t have such and such properties. And I feel that
this might be something that carries over to other notions of computations as
well, because there’s nothing intrinsic about the lambda calculus that makes
this [uniquely applicable to it].
Stephen Wolfram: Let me start by saying that I’m a very anti-type person.
And it turns out that, you know, in the history types were invented as a hack
by Russell basically to avoid certain paradoxes – and types then became this
kind of “great thing” that were used as an example and then as a practical
matter of engineering in the early computer languages there was this notion of
40
integer types versus real types and so on, and the very idea of types became
very inflated – at least that’s how I see it.
So, for example in Mathematica there are no types. It’s a symbolic system
where there is only one type: a symbolic expression. And in practical computing
the most convincing use of types is the various kinds of checking but in a sense
when something is checkable using types that involves a certain kind of rigidity
in programs that you can write, that kind of restricts the expressivity of the
language that you have. And what we found over and over again in Mathematica, as we thought about putting things in that are like types, that to do that
would effectively remove the possibility of all sorts of between paradigm kinds
of programming, so to speak, that exist when you don’t really have types.
So, having said that, this question of the analogy between proof processes
and computation processes is an interesting one. I’ve thought about that a lot,
and there’s more than just a few things to say about it. But one thing to think
about is: “What is a proof?” and “What’s the point of doing a proof?” I mean,
the real role of a proof is as a way to convince (humans, basically) that something
is true. Because when we do a computation, in the computation we just follow
through certain steps of the computation and assuming that our computer is
working correctly the result will come out according to the particular rules that
were given for the computation. The point of a proof is somehow to be able
to say to a human – look at this: you can see what all the steps were and
you can verify that it’s correct. I think the role of proofs in modern times has
become, at best, a little bizarre. Because, for example, so here’s a typical case
of this: when Mathematica first existed twenty years ago one would run into
mathematicians who would say “How can I possibly use this, I can’t prove that
any of the results that are coming out are correct!” OK? That’s what they
were concerned about. So, I would point out sometimes that actually when you
think you have a proof, in some journal for example, it’s been maybe checked
by one person – maybe – if you’re lucky. In Mathematica we can automate
the checking of many things and we can do automatic quality assurance, and
it’s a general rule that – in terms of how much you should trust things – the
more people use the thing that you’re using the more likely it is that any bugs
in it will have been found [by the time you use it]. So, you know, if you say:
“Well, maybe it’s a problem in the software that I myself am writing, maybe
it’s a problem in the system (like Mathematica) that I am using, maybe it’s
a problem in the underlying hardware of the computer” – it gets less and less
plausible that there’s a problem, the broader the use of the thing is.
So I think as a practical matter when people say: “I want a proof!” that the
demand for proof, at least in the kind of things that Mathematica does, decayed
dramatically in the first few years that Mathematica existed because it became
clear that most likely point of failure is where you as a human were trying to
explain to the computer what to do. Now, having said that it’s interesting [that]
in Mathematica we have more and more types of things that are essentially proof
systems – various kinds of things, for example proof systems for real algebra,
we just added in the great new Mathematica 7.0 all sorts of stuff of doing
computation with hundreds of thousands of variables and so on. Those are
41
effectively places where what we’ve done was to add a proof system for those
kinds of things. We also added a general equational logic proof system, but again
I think that this question whether people find a proof interesting or whether
they just want the results – it seems that the demand for presentation of proofs
is very low.
Tom Toffoli (wants to add something, Wolfram uses the break to drink some
water): If you would give me the microphone for one second: coming back
to Russell when he published Principia Mathematica – most of the theorems
were right, but a good fraction of the proofs were found wrong. I mean this
was Russell, OK? He was wrong, but there was no problem, because he was still
convinced, by his own ways he was convinced of the theorems. So he put together
some proofs (arguments) to try to convince the readers that the theorems were
right, and he convinced them. But the proofs, as actual mechanical devices,
were not working. So his proofs were coming out of just a heuristic device and
he derived, and you can always derive the right theorem with the wrong proof.
What are you going to do in that case? [Wolfram makes a sign that he can take
the the microphone back, Toffoli returns it]
Stephen Wofram: No, I think it’s actually interesting this whole analogy between proof and computation and so on. One of the things that I have often
noticed is that if you look at people’s earlier attempts to formalize mathematics
– the thing that they focused on formalizing was the process of doing proofs, and
that was what Whitehead, Russell, Peano before him and so on [worked on]. It
turned out that that direction of formalization was fairly arid. Not much came
from it. The direction that turned out to be the most interesting direction of
formalization was, in fact, the formalization of the process of computation. So,
you know, in the construction of Mathematica, what we were trying to do was
to formalize the process of computing things. Lots of people used that and did
various interesting things with it. The ratio of people who do computation with
formalized mathematics to the number who do proofs with formalized mathematics is a huge ratio. The proof side turned out not to be that interesting. A
similar kind of thing, and an interesting question, is how one would go about
formalizing every day discourse: one can take (the) everyday language and one
can come up with a formalized version of it that expresses things in a sort of
formal, symbolic way. But the thing that I’ve not figured out – actually I think
that I’ve now figured it out, but I hadn’t figured it out – was “So, what’s the
point of doing that?” In other words the Russell-Whitehead effort of formalizing
proofs turned out not not lead to much. The right idea about formalizing mathematics was the one about formalizing the process of computation. Similarly
formalizing everyday discourse as a way to make the semantic web or some such
other thing, probably has the same kind of issue as the kind of formalization of
proof as was done in mathematics and I think that maybe there is another path
for what happens when you formalize everyday discourse, and it’s an interesting
analogy to what would happen in the mathematics case. You know: what’s the
point of formalization and what can you do with a formalized system like that.
42
Gregory Chaitin: Let me restate this very deep remark that Stephen has just
made about proofs versus computation: if you look at a[ny] formal system for
a mathematical formal theory, Gödel shows in 1931 that it will always be incomplete. So any artificial language for doing mathematics will be incomplete,
will never be universal, will never have every possible mathematical argument.
There is no formal language for mathematics where every possible mathematical
argument or proof can be written in. Zermelo-Fraenkel set theory, you know,
as a corollary of Gödel – may be wonderful for everything we have but it is
incomplete. Now the exact opposite – the terminology is different – when you
talk about a programming language you don’t talk about completeness and incompleteness, you talk about universality. And the amazing thing is [that] the
drive for formalism, and Russell-Whitehead is one data point on that, another
data point is Hilbert’s program, the quest for formalization started off in mathematics, and the idea was to formalize reasoning; and the amazing thing is that
this failed. Gödel in 1931 and Turing in 1936 showed that there are fundamental
obstacles– it can’t work! But the amazing thing is that this is a wonderful failure! I mean what can be formalized beautifully, is not proof, or reasoning but:
computation. And there almost any language you come up with is universal,
which is to say: complete – because every algorithm can be expressed in it. So
this is the way I put what Stephen was saying.
So Hilbert’s dream, and Russell and Whitehead failed gloriously – is not good
for reasoning but it’s good as a technology, is another way to put it, if you’re
trying to shock people. The quest for a firm foundation for mathematics failed,
but gave rise to a trillion dollar industry. (Toffoli asks for the microphone)
Tom Toffoli: Let me add something to this. You’ve heard of Parkinson’s law26 ,
the one that says: “a system will use as many resources as are available.” It was
formulated because it was noticed that the whole British Empire was run by
essentially a basement of a few dozen people for two hundred years. And then,
the moment the British started losing their empire, they started de-colonizing
and so on, then they had a ministry of the colonies and this ministry grew bigger
and bigger and bigger as the colonies became fewer and fewer and fewer. And
it was not working as well as before. And I think that formalization is often
something like that. You can think about it, but you don’t want to actually do
it, even von Neumann, you know, the moment he decided that CAs were, sort
of, plausible to give life – he didn’t go through the process of [developing] the
whole thing. The point was already made.
Gerardo Ortiz: So, there is room for a last question ...
New question27 from the audience: If I may, can I add something about the
Curry-Howard isomorphism before I ask my question? Yes? Maybe short ... I
think that the revolution that we are living in physics is only part of a wider
revolution and there are many questions in physics to which the answer uses
the notion of algorithm. For instance: what are the laws of nature? They are
26 http://en.wikipedia.org/wiki/Parkinson’s
27 Hector
may be able to identify him.
Law
43
algorithms. So you may give this answer, only because you have the notion of
algorithm. And for centuries we didn’t have it. So, to these questions we had
either no answer or ad-hoc answers. For instance: what are the laws of nature?
Compositions. When we had compositions we didn’t have algorithms. It was
a good way to answer it. And there are many many areas in knowledge where
there are many questions to which now we answer: it is an algorithm. And one
of the very first questions on which we changed our mind, was the question:
what is a proof? And: what is a proof? The original answer was a sequence
of formulas verifying deduction rules and so on. And starting with Kolmogorov
– because behind the Curry-Howard isomorphism there is [the] Kolmogorov
interpretation – is this idea that proofs, like the laws of nature are (in fact)
algorithms. So they are two facets, or two elements of a wider revolution, and
I think that they are connected in this way.
Now consider this question28 : Does the Higgs boson29 exist? Today, I guess,
there are people who believe that the Higgs boson exists, there are people who
believe that it doesn’t exist, but anyone – you can take the electron, if you want
(instead of the Higgs boson) [which] most people I guess, believe it exists – but I
guess that everyone agrees that we have to find some kind of procedural [means
of verifying such a prediction]. It can be an experiment, it may be anything you
want – in this case – that will allow eventually, if we are lucky enough, to solve
this question. And I would be very uncomfortable if someone told me that the
Higgs boson exists in the eye of the beholder, or that Higgs boson has to exist
because then the theory would be more beautiful, or if the Higgs boson does
not exist then it’s actually a problem of the Universe and not ours and we can
continue to postulate that it exists because it would be nicer. So I wonder if the
two questions we have (not?) been discussing today are not of the same kind:
we should try to look for a kind of procedure to answer the question at some
point. One: is the universe computable? The other: is the universe discrete? In
some sense are these questions only metaphysical questions or are they questions
related to experiments that we could [and should, in fact] carry out?
George Johnson: Well thank you, and I’m sorry we won’t going to have time
to get into that, which could easily be another hour because I’m told by Adrian
that we’re on a very very draconian time schedule here because of our lunch
and also because we have to check out of our rooms. So I want to thank the
speakers very much very much for a wonderfully ...
Stephen Wolfram: Can we respond to that?
George Johnson: Perhaps we could resume at the next conference and this
would be another reason to look forward to it.
Stephen Wolfram: So many things can happen between now and then – by
then we may even have a definitive answer. Right now there’s so much to say
about it but it sounds like we don’t have the time for that.
28 George
Johnson says: “Well, thank you!” for the first time.
boson
29 http://en.wikipedia.org/wiki/Higgs
44
Everybody is in good spirits and the round table debate, along with the conference, ends (applause etc.)
End of transcript.
Acknowledgments
Transcript by Adrian German (Indiana University) with some assistance from
Anthony Michael (UC Berkeley).
References
No time to make a list of references, just some footnotes in the text.